etude du modèle d'agrégation limitée par diffusion ...régularité que peut atteindre un tel...

UNIVERSITÉ DE PARIS OUEST NANTERRE LA DÉFENSE

ÉCOLE DOCTORALE CONNAISSANCE,LANGAGE, MODÉLISATION - ED 139

MATHÉMATIQUES

T H È S Epour obtenir le titre de

Docteur en Sciences

de l'Université de Paris Ouest Nanterre La Défense

Mention : Mathématiques

Présentée et soutenue par

Cyrille Lucas

Étude du modèle de l'AgrégationLimitée par Diusion Interne

Thèse dirigée par Nathanaël Enriquez

Étude du modèle de l'Agrégation Limitée par Diusion Interne

Résumé : Cette thèse contient quatre travaux sur le modèle d'Agrégation Limitéepar Diusion Interne (iDLA), qui est un modèle de croissance pour la constructionrécursive d'ensembles aléatoires.

Le premier travail concerne la dimension 1 et étudie le cas où les marchesaléatoires formant l'agrégat évoluent dans un milieu aléatoire. L'agrégat normaliséconverge alors non pas vers une forme limite déterministe comme dans le cas demarches aléatoires simples mais converge en loi vers un segment contenant l'originedont les extrémités suivent la loi de l'Arcsinus.

Dans le deuxième travail, on considère le cas où l'agrégat est formé par desmarches aléatoires simples en dimension d > 2. On donne alors des résultats deconvergence et de uctuations sur la fonction odomètre introduite par Levine etPeres, qui compte en chaque point le nombre de passages des marches ayant formél'agrégat.

Dans le troisième travail, on s'intéresse au cas où l'agrégat est formé par desmarches aléatoires multidimensionnelles qui ne sont pas centrées. On montre quesous une normalisation appropriée, l'agrégat converge vers une forme limite quis'identie à une vraie boule de chaleur. Nous répondons ainsi à une question ouverteen analyse concernant l'existence d'une telle boule bornée.

Le quatrième travail concerne le cas particulier où une borne intérieure estconnue pour l'agrégat. On donne alors des conditions susantes sur le graphe ainsique sur la nature de cette borne pour qu'elle implique une borne extérieure. Cerésultat est appliqué au cas de marches évoluant sur un amas de percolation pararêtes surcritique, complétant ainsi un résultat de Shellef.

Mots-clés : Marche aléatoire, Modèle de croissance, Théorie du potentiel para-bolique, Percolation, DLA Interne, Tas de sable divisible.

On the Internal Diusion Limited Aggregation model

Abstract : This thesis contains four works on the Internal Diusion LimitedAggregation model (iDLA), which is a growth model that recursively builds randomsets.

The rst work is set in dimension 1 and studies the case where the ran-dom walks that build the aggregate evolve in a random environment. Thenormalised aggregate then does not converges towards a deterministic limitingshape as it is the case for simple random walks, but converges in law towardsa segment that contains the origin and which extremal points follow the Arcsine law.

In the second work, we consider the case where the aggregate is built by simplerandom walks in dimension d > 2. We give convergence and uctuation results onthe odometer function introduced by Levine and Peres, which counts at each pointthe number of visits of walkers throughout the construction of the aggregate.

In the third work, we examine the case where the aggregate is built using mul-tidimensional drifted random walks. We show that under a suitable normalisation,the aggregate converges towards a limiting shape which is identied as a trueheat ball. We thus give an answer to an open question in analysis concerning theexistence of such a bounded shape.

The last work deals with the special case where an interior bound is known forthe aggregate. We give a set of conditions on the graph and on the nature of thisinterior bound that are sucient to imply an outer bound. This result is appliedto the case of random walks on the supercritical bond percolation cluster, thuscompleting a result by Shellef.

Keywords : Random walk, Growth model, Parabolic potential theory, Perco-lation, Internal DLA, divisible sandpile.

Thèse préparée au laboratoire MODAL'X,Université de Paris Ouest Nanterre La Défense,

Bâtiment G, bureau E04200 avenue de la République,

92000 Nanterre

Table des matières

1 Introduction 11.1 Le modèle d'agrégation limitée par diusion interne . . . . . . . . . . 1

1.1.1 Motivations et applications . . . . . . . . . . . . . . . . . . . 11.1.2 Historique du modèle . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Quelques résultats sur le modèle de l'iDLA . . . . . . . . . . . . . . . 31.2.1 Convergence de l'iDLA pour les marches aléatoires simples . . 41.2.2 Amélioration de la précision . . . . . . . . . . . . . . . . . . . 51.2.3 Marches aléatoires générales . . . . . . . . . . . . . . . . . . . 61.2.4 Modèles reliés . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Marches aléatoires en milieu aléatoire . . . . . . . . . . . . . . . . . . 81.3.1 RWRE en dimension 1 . . . . . . . . . . . . . . . . . . . . . . 91.3.2 Marches aléatoires sur l'amas de percolation par arêtes surcri-

tique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Présentation des résultats . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 iDLA dans l'environnement de Sinai . . . . . . . . . . . . . . 121.4.2 Convergence et uctuations de l'odomètre . . . . . . . . . . . 141.4.3 La forme limite de l'iDLA non centré : une vraie boule de

chaleur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4.4 A la recherche d'une borne supérieure pour l'iDLA . . . . . . 22

Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 The Arcsine law as the limit of the internal DLA cluster generatedby Sinai's walk 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Notations and main results . . . . . . . . . . . . . . . . . . . . . . . 282.3 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.1 Good environments . . . . . . . . . . . . . . . . . . . . . . . . 302.3.2 The quenched localization of dn . . . . . . . . . . . . . . . . . 332.3.3 Characterization of the law of d∗ . . . . . . . . . . . . . . . . 35

2.4 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Asymptotics and uctuations of the odometer function for theiDLA model 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Denitions, main results . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.6 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

iv Table des matières

3.7 Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.8 Back to a result of Lawler, Bramson and Grieath . . . . . . . . . . 64Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 The Limiting Shape for Drifted Internal Diusion Limited Aggre-gation is a True Heat Ball 674.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3 Unfair divisible sandpile . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.1 Denitions and notations . . . . . . . . . . . . . . . . . . . . 704.3.2 Convergence and abelian property . . . . . . . . . . . . . . . 724.3.3 Limiting shape . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3.4 Regularity of D . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4 Drifted iDLA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.4.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.4.2 Limiting shape . . . . . . . . . . . . . . . . . . . . . . . . . . 884.4.3 A more natural class of drifted random walks . . . . . . . . . 98

4.5 Properties of the cluster . . . . . . . . . . . . . . . . . . . . . . . . . 994.5.1 Bounds on the cluster . . . . . . . . . . . . . . . . . . . . . . 1004.5.2 Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Containing internal diusion limited aggregation 1055.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.1.1 Denition of the iDLA aggregate . . . . . . . . . . . . . . . . 1075.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.1.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Supercritical percolation . . . . . . . . . . . . . . . . . . . . . . . . . 1095.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A Algebraic properties of the divisible sandpile model 115A.1 Lipschitz regularity of the divisible sandpile model . . . . . . . . . . 115A.2 Associativity and linear property of the divisible sandpile model . . . 116

Chapitre 1

Introduction

Contents1.1 Le modèle d'agrégation limitée par diusion interne . . . . 1

1.1.1 Motivations et applications . . . . . . . . . . . . . . . . . . . 1

1.1.2 Historique du modèle . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Quelques résultats sur le modèle de l'iDLA . . . . . . . . . . 3

1.2.1 Convergence de l'iDLA pour les marches aléatoires simples . 4

1.2.2 Amélioration de la précision . . . . . . . . . . . . . . . . . . . 5

1.2.3 Marches aléatoires générales . . . . . . . . . . . . . . . . . . . 6

1.2.4 Modèles reliés . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Marches aléatoires en milieu aléatoire . . . . . . . . . . . . . 8

1.3.1 RWRE en dimension 1 . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Marches aléatoires sur l'amas de percolation par arêtes surcri-

tique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Présentation des résultats . . . . . . . . . . . . . . . . . . . . 12

1.4.1 iDLA dans l'environnement de Sinai . . . . . . . . . . . . . . 12

1.4.2 Convergence et uctuations de l'odomètre . . . . . . . . . . . 14

1.4.3 La forme limite de l'iDLA non centré : une vraie boule de chaleur 19

1.4.4 A la recherche d'une borne supérieure pour l'iDLA . . . . . . 22

Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.1 Le modèle d'agrégation limitée par diusion interne

1.1.1 Motivations et applications

Le modèle d'agrégation limitée par diusion interne, ou iDLA (pour internalDiusion Limited Aggregation) est un modèle mathématique qui présente un algo-rithme pour la construction récursive et aléatoire d'un agrégat sur un graphe. Onlance une particule à l'origine, et on la laisse évoluer aléatoirement dans l'agrégatjusqu'à ce qu'elle en sorte. Le premier site visité à l'extérieur de l'agrégat est alorsajouté à l'agrégat, et l'on dira que la particule l'occupe. On recommence ensuite enlançant une nouvelle particule à l'origine.

Lorsque le nombre de particules lancées croît, elles donnent naissance à un agré-gat dont la taille grossit, et dont les propriétés à grande échelle dépendent de nom-breux facteurs, comme la nature du graphe ou la loi de la marche aléatoire utilisée.

2 Chapitre 1. Introduction

L'étude du modèle de l'iDLA s'intéresse donc aux propriétés de l'agrégat, de saformation et de sa dépendance en les diverses données du problème.

Cette étude est motivée par des problèmes concrets ; c'est en eet dans un articlede chimie, publié par Meakin et Deutch en 1986, que le modèle apparaît pour la pre-mière fois ([18]). L'iDLA est alors présenté comme un outil permettant de pratiquerl'electropolishing, c'est à dire l'amélioration de la régularité de la surface d'un métal.On procède par réaction d'oxydo-réduction, en faisant du métal l'anode dans unesolution appropriée. En attaquant les atomes de la structure du métal, les oxydantsen solution produisent un "agrégat" de sites vides. L'intérêt du procédé est que cetagrégat est lisse, et les auteurs, qui présentent le résultat sous forme qualitative,notent que "it is also of some fundamental signicance to know just how smooth

a surface formed by diusion limited processes may be" (la question du degré derégularité que peut atteindre un tel agrégat est d'une importance fondamentale).De nombreux procédés industriels s'apparentent ainsi à notre modèle ; on distinguel'anodic levelling qui s'attaque aux défauts de plus de 1 micron, de l'anodic brigh-

tening qui concerne les défauts de moins de 1 micron. Une vue d'ensemble de cesméthodes est présentée dans [12].

Les applications de l'electropolishing sont multiples ; les surfaces ainsi traitéesétant très régulières, elles sont moins sensibles à la corrosion, et peuvent être net-toyées plus facilement, ce qui est par exemple nécessaire en médecine.

1.1.2 Historique du modèle

L'introduction du modèle de l'iDLA dans le domaine des mathématiques est dueà Diaconis et Fulton ([6]). Étant donnés deux ensembles nis A,B ∈ Zd, les auteursdénissent leur smash sum A ⊕ B comme un certain ensemble aléatoire dont lecardinal est la somme de ceux de A et B. On note A ∩ B = x1, · · · , xk, et l'ondénit l'ensemble de départ suivant,

C(0) = A ∪B.

On se donne ensuite une famille de marches aléatoires simples(Si)i=1···k, telles que

Si(0) = xi. On dénit récursivement les temps d'arrêt et les ensembles suivants :

νi = inft > 0 : Si(t) 6∈ C(j − 1), etC(i) = C(i− 1) ∪ Si(νi).

A⊕B = C(k)

L'observation clé qui justie cette dénition est que la loi de A⊕B ne dépend pasde l'ordre choisi pour indexer A ∩ B. Cette propriété est une conséquence de lapropriété de Markov forte, appliquée aux marches Si et aux temps d'arrêts νi. Demême, la propriété de Markov implique l'associativité de cette addition :

A⊕ (B ⊕ C) = (A⊕B)⊕ C.

1.2. Quelques résultats sur le modèle de l'iDLA 3

Ainsi, on peut sans diculté multiplier un ensemble par un nombre entier n > 0,en l'additionnant n fois avec lui-même. C'est l'étude de ce cas particulier qui donneranaissance au modèle d'iDLA proprement dit ; on considère le singleton origine, eton l'additionne n fois avec lui-même. Le problème est alors de savoir quelles sont lespropriétés asymptotiques du résultat de l'addition.

Cette dénition est reprise dans le premier article consacré au modèle par Law-ler, Bramson et Grieath en 1992, [15]. An de dénir directement le modèle, onconsidère (Si)i∈N une suite de marches aléatoires simples indépendantes sur Zd,telles que Si(0) = 0.

L'agrégat d'iDLA A(n) et les temps d'arrêts νi sont construits simultanémentde la façon suivante :

ν1 = 0,

A(1) = 0 = S1(ν1), et∀i > 1, νi = inft > 0 : Si(t) 6∈ A(i− 1),

A(i) = A(i− 1) ∪ Si(νi).

Dans le cas unidimensionnel, Diaconis et Fulton prouvent la convergence presquesûre de l'agrégat renormalisé vers le segment [−1

2 ,12 ]. Pour les dimensions supé-

rieures, Lawler, Bramson et Grieath ont prouvé la convergence presque sûre, ausens de Hausdor, de l'agrégat renormalisé vers la boule unité euclidienne. Nousdonnerons un résumé de ces résultats dans la suite de cette introduction.

Figure 1.1 Agrégat sur Z2 (remormalisé) après 15, 122 et 1295 itérations respec-tivement

1.2 Quelques résultats sur le modèle de l'iDLA

Dans cette partie, nous présentons les principaux résultats sur le modèle del'iDLA, ainsi que l'historique des arguments les plus classiques


1.2.1 Convergence de l'iDLA pour les marches aléatoires simples

Le cas des marches aléatoires simples a été le premier à être étudié ; dès l'articlede Diaconis et Fulton, la convergence presque sûre de l'agrégat renormalisé est dé-montrée dans le cas particulier de la dimension 1. Ce théorème est ensuite généralisédans l'article de Lawler, Bramson et Grieath ([15]) aux dimensions supérieures.

Soit A(n) l'agrégat au rang n, construit avec des marches aléatoires simplesdans Zd. Notons Bk la boule euclidienne de centre 0 et de rayon k dans Zd. Alorsle résultat de [15] s'exprime de la façon suivante : pour tout ε > 0, on a presquesûrement, à partir d'un certain rang,

Bn(1−ε) ⊂ A (|Bn|) ⊂ B(n(1+ε).

Même s'il a depuis été très largement amélioré, ce premier résultat, ainsi que lesméthodes utilisées pour l'obtenir, constituent la base de la plupart des travaux sur lemodèle d'iDLA. En eet, les auteurs utilisent deux techniques qui seront largementréutilisées par la suite, et en particulier dans certaines des preuves des chapitres quisuivent.

La première de ces techniques est une extension de la remarque de Diaconis etFulton : la loi d'un agrégat ne dépend pas de l'ordre dans lequel les marches sontlancées. Les auteurs remarquent que l'on peut même aller plus loin : on peut lancerune marche donnée, l'arrêter à un temps d'arrêt, garder en mémoire sa position, et larelancer ultérieurement, après d'autres marches par exemple. Ainsi, pour la preuvede la première inclusion Bn(1−ε) ⊂ A (|Bn|), on construit un agrégat A ⊂ Bn(1−ε)en utilisant la règle d'agrégation classique, modiée de la façon suivante : si uneparticule sort de Bn(1−ε) avant d'ajouter à l'agrégat, on l'arrête et on retient saposition. Une nouvelle marche est alors lancée de l'origine. Après avoir lancé nmarches, on obtient un couple formé d'un agrégat intermédiaire A, et d'un ensemblede positions où des marches sont arrêtées. L'agrégat nal est obtenu en relançant lesmarches arrêtées, et en les laissant cette fois évoluer jusqu'à leur sortie de l'agrégat,suivant la règle d'agrégation normale. L'agrégat obtenu au nal contient A, et saloi est la même que celle de A. Si l'on prouve que Bn(1−ε) ⊂ A (|Bn|) , le résultatrestera valide pour l'agrégat.

La seconde technique développée par les auteurs est l'introduction des sommesd'indicatrices suivantes, dénies pour un point z ∈ Zd.

N(n,m, z) : Nombre de marches qui visitent z avant d'ajouter à l'agrégat

ou de sortir de Bn, parmi les m premières.

M(n,m, z) : Nombre de marches qui visitent z avant de sortir de Bn,parmi les m premières.

L(n,m, z) : Nombre de marches qui visitent z après avoir ajouté à l'agrégat,

mais avant de sortir de Bn, parmi les m premières.

La variable aléatoire N mesure le nombre de marches qui passent en z avant leur


sortie de l'agrégat. Si N > 0 en un point z, cela signie que des marches y sontpassées avant leur sortie de l'agrégat, donc que z est un point de l'agrégat A(n).

Les deux autres variables aléatoires sont utilisées pour estimer N ; en eet, onremarque que M = N + L, et que M ne dépend pas du tout de l'agrégat. Quant àL, s'il dépend a priori de l'agrégat, on peut en fait le majorer stochastiquement parune somme de variables indépendantes, en procédant comme suit.

La variable L est une somme, pour toutes les marches partant de l'origine, del'indicatrice de l'évènement suivant : elles ajoutent d'abord à l'agrégat, puis visitentz. On peut indexer ces variables par le point auquel elles ajoutent à l'agrégat ; eneet, il y a un seul tel point par marche, et une seule marche (au plus) pour chaquepoint. Au lieu de sommer sur les indices des marches, on peut donc sommer surles points de Bn les évènement correspondant au fait que ces marches passent parz avant de sortir de Bn. On remarquera que les variables aléatoires qui constituentcette somme sont maintenant indépendantes entre elles.

Cette écriture est la base de nombreuses preuves, et son utilisation est omnipré-sente dans le cadre de l'étude de notre modèle.

1.2.2 Amélioration de la précision

Pour étudier la précision de la convergence de l'agrégat vers sa forme limite,il est classique d'introduire les variables aléatoires qui mesurent les erreurs interne(inner) et externe (outer) respectivement :

δI(n) = n− infz 6∈A(|Bn|)

||z||,

δO(n) = supz∈A(|Bn|)

||z|| − n.

Majorer ces erreurs permet d'obtenir une plus grande précision sur la vitessede convergence de l'agrégat. Ainsi, Lawler prouve en 1995 ([13]) que l'erreur estsous-diusive ; plus précisément, on a, pour d > 2,

δI(n) = o(n1/3(lnn)2

), et

δO(n) = o(n1/3(lnn)4

).

Les preuves de ces résultats sont très similaires à celles employées dans [15] ; ellesreposent principalement sur les estimées précises de la fonction de Green arrêtéeobtenues par Lawler dans [14].

Ces résultats ont été à nouveau drastiquement améliorés récemment, grâce auxtravaux simultanés de Asselah et Gaudillère ([1], [2]) d'une part, et de Jerison, Levineet Sheeld ([9], [10], [11] ) d'autre part. Les uctuations de l'agrégat autour de saforme limite sont à nouveau majorées : en dimension 2, on a :

δI(n) = O (lnn) , et

δO(n) = O (lnn) .


Quand d > 2, la borne est encore plus forte, puisque

δI(n) = O(√

lnn), et

δO(n) = O(√

lnn).

Ces résultats conrment les conjectures dues aux simulations, qui semblaientindiquer une croissance seulement logarithmique de l'erreur.

Remarquons que jusque là, nous n'avons parlé que de majorant pour les erreurs δIet δO. Leur ordre de grandeur exact est encore inconnu, puisque nous ne connaissonsque des bornes supérieures. (Ainsi, l'identication des uctuations moyennes parJerison, Levine et Sheeld, [11], ne concerne qu'un version moyennée des erreurs δIet δO, et ne fournit donc pas de borne inférieure à proprement parler.)

1.2.3 Marches aléatoires générales

Tous les résultats cités ci dessus concernent les marches aléatoires simples ; laquestion naturelle qui se pose alors est de savoir si cette convergence vers une formelimite s'étend à d'autres marches aléatoires, et si oui, sous quelles conditions et versquelle forme limite.

Dans son article [3], Blachère présente un résultat de convergence pour desmarches aléatoires centrées, sous des conditions de moment. La convergence a bienlieu, mais la forme limite dépend de la loi des marches aléatoires ; il s'agit en fait d'unellipsoïde, d'autant plus allongée dans une direction que la variance de la marchealéatoire dans cette direction est grande.

Considérons une marche aléatoire S, partant de l'origine. Supposons S centréeet irréductible (pour avoir un bon recouvrement). On dénit :

p(x) = P(S1 = x), et

q(θ) =∑x∈Zd

(θ.x)2p(x).

Soit Q la matrice de covariance de la forme quadratique q, et on dénit la normeassociée à cette matrice dénie positive ;

||x||Q = (d−1txQ−1x)1/2,

Alors, sous des conditions de moment, on a presque sûrement convergence de l'agré-gat renormalisé vers la boule unité associée à la norme || · ||Q.

Dans une seconde partie, Blachère s'intéresse au cas des marches aléatoires noncentrées. Le cas de la dimension 1 est traité, et la convergence vers le segmentunité est montrée, avec un degré de précision logarithmique. L'auteur pose ensuitela question de l'existence d'une forme limite pour l'agrégat renormalisé, après avoirrejeté la conjecture initiale que la forme limite correspondrait à une ligne de niveaude la fonction de Green. Nous apporterons une réponse précise à cette question au


chapitre 4.

1.2.4 Modèles reliés

Levine et Peres montrent, dans leur article [17], que l'étude du modèle de l'iDLAest reliée à celle de deux autres modèles. Le premier de ces modèles est le modèledu rotor router. Le principe est de construire un agrégat, en utilisant, en lieu etplace des marches aléatoires, des "marcheurs eulériens". Cet analogue déterministedes marches aléatoires a été introduit par Priezzhev et al. [19]. En chaque site deZ2 est placé un rotor, qui pointe vers le Nord, Sud, Est ou Ouest. Un marcheurse déplace sur Z2 selon la règle suivante : à chaque étape, le rotor du site occupépar la particule est tourné d'un angle droit dans le sens des aiguilles d'une montre.Puis le marcheur se déplace vers le site voisin dans la nouvelle direction indiquéepar le rotor. En dimension supérieure, on dénit le modèle de façon similaire, en sedonnant un ordre sur les 2d directions possibles pour le rotor. Les rotors commencentdans une position arbitraire, et on construit l'agrégat en conservant les positionsintermédiaires des rotors.

Dans le modèle du divisible sandpile, on utilise au lieu des particules une quan-tité de masse continue. Une conguration de masse sur Zd est une fonction positive,bornée et à support compact, qui à chaque site chaque site associe la masse qu'ilcontient. Un site est plein si ça masse est supérieure ou égale à 1. Il peut alorss'écrouler, c'est à dire répartir son excès de masse équitablement entre ses voisins,et garder une masse égale à 1. On part d'une conguration de masse, et on réaliseà chaque étape de temps un écroulement sur un des sites pleins. Lorsque le nombrede ces écroulements tend vers l'inni, la conguration de masse tend vers une con-guration nale, à condition que chaque site plein soit écroulé une innité de fois.Cette conguration nale, dans laquelle chaque site a une masse inférieure ou égaleà 1. Cette convergence est prouvée dans [19]. De plus, même si deux écroulementsindividuels ne commutent pas, la conguration nale ne dépend pas de l'ordre desécroulements. Nous donnons en annexe de ce travail (Annexe A) quelques résultatsoriginaux d'ordre algébrique sur la fonction DS, qui associe à une congurationinitiale sa condition nale obtenue après écroulements.

Ces deux modèles sont étudiés en détail dans [16] ; lorsque toute la masse estlancée à l'origine, l'agrégat crée par les deux modèles, une fois renormalisé, convergevers la boule euclidienne. L'article [17] s'intéresse aux formes limites de ces modèles,ainsi qu'à celle de l'iDLA, lorsque la masse est répartie selon une congurationinitiale où toute la masse n'est pas lancée de l'origine.

Pour ces trois modèles, une fonction odomètre est introduite, qui mesure la masseémise depuis un point donné lors de la construction de l'agrégat :

u(x) = masse totale émise depuis x.

Pour le divisible sandpile, cette quantité est bien dénie, et ne dépend pas, elle nonplus, de la suite d'écroulements choisie. Pour les modèles de l'iDLA et du rotor


router, on dénit l'odomètre comme le nombre de particules qui passent au point xsans s'y arrêter (ce qui correspond au nombre de particules émises par x).

Soit y l'un des voisins de s ; dans chacun des trois modèles, la masse émise par yen direction de x peut être comparée à 1

2du(y). Dans le cas du divisible sandpile, ona une égalité, qui est seulement vraie avec une erreur inférieure à 2d dans le cas dumodèle du rotor router. Dans le cas de l'iDLA, cette égalité semble vraie ; en eet,chacune des particules choisit aléatoirement la direction dans laquelle elle quittey, donc si ces variables étaient indépendantes, on pourrait avoir une telle égalitéen espérance. Ce n'est pas le cas, mais une modication adéquate de la fonctionconsidérée est utilisée pour traiter cette diculté. Ainsi, la masse totale reçue parx de la part de ses voisins est 1

2d

∑y∼x u(y). De ce fait, on a l'équation de masse

locale :∆u(x) = ν(x)− σ(x),

avec σ et ν les congurations initiales et nales de masse, respectivement, et ∆ leLaplacien discret sur Zd, déni par ∆u(x) = 1

2d

∑y∼x (u(y)− u(x)). On sait que la

masse nale vaut 1 à l'intérieur de l'agrégat, et 0 à l'extérieur, tandis qu'elle prenddes valeurs comprises entre 0 et 1 sur la frontière.

L'étude de cette équation discrète, et de son analogue continu permet de prouverla convergence des agrégats obtenus dans le cadre des trois modèles.

C'est ainsi que, sous les conditions de convergence appropriées pour la congu-ration initiale, les agrégats renormalisés pour les trois modèles convergent au sensde Hausdor, respectivement presque sûrement au sens de Hausdor, vers la mêmeforme limite.

Nous étudions la fonction odomètre de l'iDLA dans le chapitre 3, où nous nousintéressons à sa convergence, et identions ses uctuations. De plus, notre techniquepour identier la forme limite de l'agrégat d'iDLA construit avec des marches aléa-toires non centrées, dans le chapitre 4, reprend le cheminement général de [17] :après avoir introduit un analogue du divisible sandpile, nous nous intéressons àl'équation discrète que les fonctions odomètre de ces modèles satisfont, et à leuranalogue continu.

1.3 Marches aléatoires en milieu aléatoire

Si les marches aléatoires classiques permettent de modéliser de nombreux phéno-mènes physiques ou chimiques, cette modélisation suppose l'homogénéité de l'espace.Or, dans de nombreux systèmes, cette hypothèse n'est pas vériée. En particulier,dans la structure d'un métal, des impuretés sont souvent présentes ; ces défauts del'environnement sont en général loin d'être négligeables, et il est ainsi apparu né-cessaire d'introduire de nouveaux modèles qui prennent en compte l'hétérogénéitéspatiale due à ces défauts.

Nous nous intéresserons ici plus particulièrement aux marches aléatoires en mi-lieu aléatoire, introduites en 1967 par le biophysicien Chernov [4]. Nous les dési-gnerons par le terme RWRE, pour Random Walk in Random Environment. Dans le

1.3. Marches aléatoires en milieu aléatoire 9

cadre de notre étude, nous sommes amenés à considérer d'une part les RWRE endimension 1 dans l'environnement de Sinai, et d'autre part la marche aléatoire surle cluster de percolation surcritique dans Zd.

1.3.1 RWRE en dimension 1

Les marches aléatoires en milieu aléatoire à une dimension sont dénies de lafaçon suivante. On choisit d'abord une suite de variables aléatoires indépendanteset identiquement distribuées ω = (ωi)i∈Z, que l'on appelle l'environnement. Étantdonné cet environnement, on appelle marche aléatoire en milieu aléatoire la chaînede Markov (Sn)n∈N dénie par S0 = 0, et pour n > 0,

Pω (Sn+1 = k|Sn = i) =

ωi si k = i+ 1,

1− ωi si k = i− 1,

0 sinon.

Signalons que l'on distingue deux lois de probabilités diérentes. La loi de pro-babilité conditionnellement à l'environnement, notée Pω, et dénie ci-dessus, estappelée la loi quenched (ce qui veut dire "trempé", selon la terminologie prove-nant de la métallurgie). Cette loi présente l'avantage d'être markovienne, mais ellen'est pas invariante par translation. La loi annealed ("recuite") est quant à elledénie comme la moyenne de la loi quenched sur tous les environnements, c'est àdire P(·) =

∫Pω(·)µ(dω), où µ est la loi de l'environnement. Cette nouvelle loi est

invariante par translation, mais non markovienne.

Le cas qui nous intéresse est celui de l'environnement de Sinai. Nous le dénissonsde la façon suivante. On se place tout d'abord sous l'hypothèse d'ellipticité suivante :

µ(ω0 = 0) = µ(ω0 = 1) = 0,

qui est garantit l'irréductibilité. On dénit par ailleurs

ρ(i) =1− ωiωi

,

et l'on demande que Eµ(log ρ(0)) = 0, ce qui garantit la récurrence de la marche,d'après le critère donné par Solomon en 1975 ([22]). Enn, la dernière hypothèsesur l'environnement est celle qui garantit la convergence du potentiel de la marchealéatoire vers un mouvement brownien, comme nous allons le voir. On demande quelog ρ(0) ait un moment d'ordre deux ni et non nul ; c'est à dire que

0 < σ2 = Eµ(

(log ρ(0))2)<∞.

Sinai [21] montre en 1982 que dans ce cas, la marche est beaucoup plus lenteque les marches aléatoires usuelles. Plus précisément, il existe une variable aléatoire


non dégénérée et non gaussienne b∞, telle que

σ2 Sn(log n)2

loi−→ b∞.

Ce phénomène contraste avec le cas des marches aléatoires usuelles qui ont uncomportement asymptotique en

√n. La démonstration Sinai fait apparaître une

quantité qui joue un rôle important, le potentiel Vω associé à un environnementdonné ω. On le dénit comme suit pour x ∈ Z :

Vω(0) = 0

Vω(x) =x∑k=1

ln ρ(k) si x > 1,

Vω(x) = −0∑

k=x+1

ln ρ(k) si x 6 − 1.

Cette quantité, qui dépend uniquement de l'environnement, joue un rôle analogueà une énergie en physique. Le caractère sous-diusif de notre marche établi parSinai est dû à la présence de pièges. Ce sont les puits de potentiel (il s'agit defonds de vallées pour le potentiel Vω), dans lesquels la marche (Sn)n∈N reste piégéelongtemps avant de réussir à s'en échapper. Elle est ensuite piégée par un autre puitsde potentiel, dans lequel elle passe encore plus de temps, et ainsi de suite.

1.3.2 Marches aléatoires sur l'amas de percolation par arêtes sur-critique

Dans cette partie, nous nous intéressons au cas de la marche aléatoire sur l'amasde percolation par arêtes surcritique.

Commençons par dénir cet ensemble (voir la dénition donnée par Grimmettdans [7]) ; pour l'ensemble des arrêtes ex,y indexé par x, y ∈ Zd, avec |x − y| = 1,on dénit une famille de variables aléatoires de Bernoulli indépendantes et identi-quement distribuées ηe, avec P(ηe = 1) = p ∈ [0, 1]. Les arêtes e telles que ηe = 1

sont appelées arrêtes ouvertes. On appelle alors chemin ouvert de x à y un ensembled'arêtes ouvertes reliant x à y, et amas ouvert tout sous ensemble de Zd tel quedeux de ses points sont toujours reliés par un chemin ouvert. Un résultat classiquede percolation stipule qu'il existe pc ∈ (0, 1) tel que pour p > pc, il existe presquesûrement un unique amas ouvert de cardinal inni, noté C∞. De plus, la probabilitéque l'origine soit dans cet amas inni est strictement positive. On conditionneradans la suite par cet évènement an de se placer dans le cas où l'origine est re-liée à l'inni. La gure 1.2 montre l'amas de percolation par arêtes sur Z2 avec leparamètre p = 0.8.

Sur cet amas inni, on dénit une marche aléatoire de la manière suivante : àchaque étape, la marche choisit de façon équiprobable l'un des voisins auxquels elleest reliée par une arête ouverte, et s'y dirige. Cette marche a été introduite par De

1.3. Marches aléatoires en milieu aléatoire 11

Figure 1.2 Percolation par arêtes sur Z2 pour p = 0.8.

Gennes sous le nom de la "fourmi dans le labyrinthe" dans [5].

On dénit le modèle de l'iDLA sur l'amas de percolation de manière similaireaux autres modèles d'iDLA, en utilisant les marches aléatoires sur l'amas inni depercolation surcritique conditionné à contenir l'origine (pour que l'agrégat puissegrossir indéniment).

Dans son article [20], Shellef montre l'existence d'une borne inférieure pourl'agrégat qui est l'intersection de la boule euclidienne de Zd avec l'amas inni depercolation surcritique. Plus précisément, pour presque tout environnement et toutε > 0, on a presque sûrement BR(1−ε) ⊂ A(|BR|) à partir d'un certain rang.

Ce résultat est un résultat de forme limite partiel que nous complétons au cha-pitre 5. Sa preuve repose principalement sur la convergence au sens faible de lamarche aléatoire sur l'amas de percolation surcritique vers un mouvement Brow-nien, ainsi que des estimées sur les temps de sortie des boules euclidiennes. Larégularité de l'amas de percolation entre aussi en jeu, avec l'utilisation de l'inéga-lité de Harnack et des bornes sur le nombre de sites contenus dans l'intersection del'amas avec les boules euclidiennes et les boites carrées.


1.4 Présentation des résultats

Cette thèse s'organise autour de quatre principaux chapitres. Le premier est issud'un travail réalisé en collaboration avec François Simenhaus et Nathanaël Enriquez,sur le modèle l'iDLA en une dimension pour les marches aléatoires dans l'environ-nement de Sinai. Le second présente une étude précise de la fonction odomètre, desa convergence et de ses uctuations pour le modèle de l'iDLA classique construità l'aide de marches aléatoires simples. Au troisième chapitre, l'agrégat d'iDLA estétudié pour des marches aléatoires non centrées. Le quatrième et dernier chapitre,issu d'une collaboration avec Hugo Duminil, Ariel Yadin et Amir Yehudayo, traitede l'agrégat d'iDLA construit sur l'amas de percolation par arêtes surcritique dansZ2. Nous donnons ici un récapitulatif des résultats obtenus, ainsi que des méthodesutilisées pour les prouver.

1.4.1 iDLA dans l'environnement de Sinai

Dans ce chapitre, on étudie le cas de marches aléatoires qui évoluent dans un en-vironnement aléatoire non homogène. Plus précisément, on utilise des marches aléa-toires récurrentes dans un environnement appelé environnement de Sinai. Selon lespropriétés de l'environnement, l'agrégat peut devenir très asymétrique, s'éloignantainsi du cas trivial des marches aléatoires simples en dimension 1 dont l'agrégatconverge vers le segment [−1

2 ,12 ].

L'agrégat d'iDLA en dimension 1 après avoir lancé n marches est un segment.On appelle dn son extrémité droite ; comme c'est un segment de longueur n, sonextrémité gauche est située en dn − n. Nous établissons la convergence de dn

n en loivers la loi de l'arcsinus sous la mesure annealed. C'est l'objet du théorème suivant :

Théorème . 1Sous P , dn/n converge en loi vers la loi de l'arcsinus. Précisément, pour tout

couple 0 6 a 6 b 6 1,

P

(dnn∈ (a, b)

)−−−→n→∞

∫ b

a

1

π√x(1− x)

dx.

Pour prouver ce théorème, la première étape est de remarquer que notre en-vironnement peut être vu comme un potentiel dans lequel la marche évolue. Lesconditions de Sinai garantissent la convergence de ce potentiel normalisé vers unmouvement Brownien. Par ailleurs, si l'on considère une vallée du potentiel, délimi-tée d'un côté par un sommet de hauteur h1 et de l'autre par un sommet de hauteurh2, on prouve que la probabilité pour la marche de sortir de cette vallée par le som-met le plus haut des deux décroît très vite lorsque |h2−h2| grandit. Ainsi, avec uneforte probabilité, la marche sort d'une vallée de potentiel par le côté où la barrièrede potentiel est la moins haute.

Ces considérations donnent une idée de notre stratégie. D'abord, en considérantle potentiel, il faut chercher la position théorique de l'agrégat, qui sera dans une

1.4. Présentation des résultats 13

vallée de largeur plus grande que 1 contenant l'origine. De plus, il contiendra toutevallée de largeur plus petite que 1 contenant l'origine. On regarde donc la plus grandevallée de largeur inférieure à 1 contenant l'origine. Sa largeur peut être strictementplus petite que 1, auquel cas on doit comprendre le comportement de l'agrégat endehors de la vallée une fois que cette dernière est remplie. Remarquons d'abord queles deux sommets délimitant la vallée n'ont pas la même hauteur. En eet, si onconsidère les temps d'atteinte par le potentiel brownien v des niveaux y à droite età gauche de l'origine,

T+y (v) = inf t > 0, tel que v(t) > y ,T−y (v) = − sup t 6 0, tel que v(t) > y ,

alors le niveau de la plus grande vallée de largeur plus petite que 1 est déni par

y = supy > 0, tel que T+y + T−y 6 1.

Comme T+y et T−y sont deux subordinateurs indépendants, leurs temps de sauts sont

disjoints, et leur somme est un subordinateur, dont les temps de saut sont ceux deT+y et T−y . Ainsi, cette somme franchit le niveau 1 lors du saut d'un seul des deux

subordinateurs, ce qui signie que le potentiel présente un maximum local de l'undes côtés, mais pas de l'autre.

On s'attend à ce que l'agrégat dépasse de la vallée de niveau y par le côté leplus bas, c'est à dire le côté où le potentiel présente un maximum local. On dénitainsi la position théorique de l'agrégat comme la vallée de niveau y, augmentée ducôté du maximum local du potentiel par la longueur nécessaire pour atteindre unagrégat de taille unité.

Bt

1

t0

y

T−y = g∗ T+

y d∗

α

Figure 1.3 Sur cette gure, le maximum local du potentiel est atteint du côté

droit, donc la position théorique de l'agrégat occupe la vallée de niveau y et déborde

du côté droit. C'est la partie en gras de la gure.

Une fois dénie la position théorique de l'agrégat, il faut prouver qu'il sera


eectivement proche de cette position. Pour ce faire, on commence par établir uneliste de conditions contraignantes qui dénissent un bon environnement. On étudiela probabilité qu'un environnement soit bon ; elle est asymptotiquement proche de1. Sur un bon environnement, les propriétés de la marche garantissent avec uneprobabilité exponentiellement proche de 1 que l'agrégat sera à distance négligeablede sa position théorique.

Ainsi, l'agrégat est asymptotiquement proche de sa position théorique. Il resteà étudier la loi de cette position théorique en tant que fonctionnelle du mouvementBrownien vers lequel converge le potentiel. Une étude des processus T+

y et T−y permetde conclure que cette position théorique suit la loi de l'arcsinus.

De plus, la position de l'agrégat autour de l'origine est très asymétrique, commele montre le théorème suivant :

Théorème . 2P−p.s., avec probabilité 1 sous la loi Pω,

lim supn→∞

dnn

= 1, et

lim infn→∞

dnn

= 0.

La preuve de ce théorème repose intégralement sur le raisonnement du théorèmeprécédent. Lorsque l'on considère l'agrégat à des échelles de temps diérentes, saposition suit notre fonctionnelle d'un mouvement Brownien indépendant à chaqueéchelle de temps. Ainsi, il existe une échelle du temps pour laquelle la position del'agrégat est arbitrairement proche du segment [0, 1], ou du segment [−1, 0].

Ceci conclut note étude de l'agrégat d'iDLA sur l'environnement de Sinai.

1.4.2 Convergence et uctuations de l'odomètre

On a vu que l'étude de la fonction odomètre jouait un rôle important dansl'étude du modèle de l'iDLA et des modèles reliés. Cependant, la convergence del'odomètre proprement dit n'a pas été étudiée jusque là pour le modèle classiquede l'iDLA où toutes les particules sont lancées à l'origine et eectuent des marchesaléatoires simples. Dans ce chapitre, nous étudions le comportement asymptotiqueainsi que les uctuations de la fonction odomètre.

Soit z ∈ Rd un point non nul de la boule unité ouverte. On note z = (x1, · · · , xd),et zn := (bnx1c, · · · bnxdc) .

Le point zn et un point de Zd dont la renormalisation est proche de z, on étudiedonc la valeur de l'odomètre en ce point, en le dénissant comme suit.

La fonction odomètre un(z) au rang n mesure le nombre de particules qui sontpassées en zn lors de la construction de l'agrégat :

un(z) :=

bωdndc∑i=0

νi∑t=0

1Si(t)=zn ,


avec νi le temps d'arrêt correspondant à l'instant où la particule s'ajoute à l'agrégat,comme nous l'avons déni plus haut.

Le premier résultat obtenu concerne la convergence presque sûre de l'odomètrenormalisé.

Théorème . 1 Soit z un point non nul de la boule unité ouverte de Rd. Quand ntend vers l'inni,

un(z)

n2→||z||2 − 1− 2 ln(||z||) presque sûrement, si d = 2,

un(z)

n2→||z||2 +

2

(d− 2)||z||d−2− d

d− 2presque sûrement, si d > 3.

Ainsi, la fonction odomètre renormalisée converge presque sûrement sur le disqueunité privé de l'origine. La gure 1.4 présente une simulation de la diérence entreunn2 et sa limite lorsque d = 2.

Figure 1.4 Sur la gure de gauche on a représenté la fonction f(z) = −||z||2 +

1 + 2 ln(||z||), et sur la gure de droite, la somme f(z) + un(z)n2

Remarquons que ces fonctions tendent vers 0 sur le bord de la boule unité,et vers l'inni au voisinage de l'origine. An de comprendre plus précisément cecomportement, nous introduisons une normalisation diérente. soit yn une suite depoints de Zd telle que yn

n converge vers 0. On s'intéresse alors au nombre de passageau point yn, c'est à dire à la suite des valeurs prises par l'odomètre en yn

n .

Nous diérencions le cas où yn reste constant de celui où yn s'éloigne de l'origine,quand n tend vers l'inni. Notre résultat est le suivant :

Théorème . 2 Soit yn une suite de points tels que ynn tend vers 0.

Si yn = a ∈ Zd,

un(ynn

) ∼ 2n2 ln(n) presque sûrement, si d = 2,

un(ynn

) ∼ ωdndG(0, a) presque sûrement, si d > 3.


Si ||yn|| tend vers l'inni lorsque n tend vers l'inni,

un(ynn

) ∼ 4n2 ln

(n

||yn||

)presque sûrement, si d = 2,

un(ynn

) ∼ 2n2

d− 2

(n

||yn||

)d−2

presque sûrement, si d > 3.

De façon similaire, on étudie le comportement au voisinage du bord en introdui-sant une nouvelle normalisation. Lorsque l'on considère un point qui se rapprochedu bord, on a le résultat suivant :

Théorème . 3 Soit yn une suite de points de Zd telle que ynn → 1, et on note ηn =

n−||yn||. On suppose qu'il existe 0 < α < β < 1 et c ∈ R∗+ tels que 1cn

α < ηn < cnβ.

Alors, les convergences suivantes ont lieu dans L2 :

un(ynn )

η2n

→ 1 si d = 2,

un(ynn )

η2n

→ 2d si d > 3.

De plus, si α > 12 , ces convergences ont lieu presque sûrement.

Les preuves de ces trois théorèmes sont reposent sur l'encadrement de l'agrégatentre deux boules euclidiennes. En eet, on a

Bn−ηI(n) ⊂ A(⌊ωdn

d⌋) ⊂ Bn+ηO(n),

ce qui entraîne une inégalité similaire sur les temps de sortie des boules euclidienneset de l'agrégat. En eet, lorsque l'on considère la j-ème marche, avec

⌊ωdk

d⌋6 j <⌊

ωd(k + 1)d⌋, on a :

ξj(k−1)−ηI(k) 6 σj 6 ξjk+ηO(k), (1.4.1)

où ξji est le temps de sortie, pour la j-ème marche, de la boule Bi.

Plaçons-nous dans le cas du théorème 1. Le nombre de passage en un point zdonné de la marche avant sa sortie de l'agrégat peut être comparé avec son nombrede passage avant sa sortie de deux boules euclidiennes asymptotiquement proches.En utilisant le résultat obtenu par Asselah et Gaudillière (see [1], [2]) qui majore ladistance entre ces deux boules euclidiennes, on obtient une approximation précisede l'odomètre en termes des deux fonctions suivantes :


un(z)− =n−1∑k=0

bωd(k+1)dc∑i=bωdkdc

ξk−(ln k)2∑t=1

1Si(t)=zn

un(z)+ =

n−1∑k=0


ξjk+1+ln(k+1)2∑

t=1

1Si(t)=zn

On étudie ensuite ces variables aléatoires. Pour ce faire, on introduit la famillede variables aléatoires :

χjk,n(z) =

ξjk∑t=1

1Sj(t)=zn ,

dont la loi est le produit d'une variable de Bernoulli par une variable géométrique.Les espérances de ces variables aléatoires sont les fonctions de Green arrêtées aubord de la boule Bk :

Gk(0, zn) = E(χjk,n(z)

).

Les estimées obtenues par Lawler ([14]) permettent l'évaluation précise de cesespérances :

Gn(0, z) =2

ω2ln

n

||z||+O(

1

||z||) +O(

1

n) si d = 2, et

Gn(0, z) =2

d− 2

1

ωd(||z||2−d − n2−d) +O(||z||1−d).

Ainsi, les espérances de u−n et u+n s'expriment comme des sommes de Riemann :

1

n2E(un(z)±) →

∫ 1

||z||4t ln(

t

||z||)dt = ϕ2(z), et

1

n2E(un(z)±) →

∫ 1

||z||

2d

d− 2(td−1||z||2−d − t)dt = ϕd(z)

Une étude détaillée des variances de u−n et u+n montre que cette convergence a

en fait lieu presque sûrement. Ainsi, ces convergences entraînent celle de unn2 . Les

preuves des théorèmes 2 et 3 sont similaires à la première preuve.

Les théorèmes qui suivent caractérisent, quand d > 3, et pour les points à dis-tance non macroscopique de l'origine, les uctuations de l'odomètre autour de savaleur moyenne. Pour d = 2 ainsi que pour les points à distance macroscopique del'origine, les termes d'erreur de notre méthode dépassent en ordre de grandeur lesuctuations, et cette dernière ne nous permet donc pas d'identier les uctuationsde l'odomètre.

Pour les points situés à distance nie de l'origine, les uctuations de l'odomètre


suivent la loi suivante :

Théorème . 4 Soit a ∈ Zd, d > 3. Lorsque n tend vers l'inni, les uctuations de

l'odomètre en a sont gaussiennes. Plus précisément, si a ∈ Zd, on dénit la fonction

et la variable aléatoire suivantes :

ψd(a) = G(0, a)G(0, 0)

(3− P(τa <∞)− 1

G(0, 0)

),

F0,n(a) = n−d/2(un

(an

)− E

(un

(an

))), if d>2.

Alors F0,n converge en loi vers un champ gaussien sur Zd, dont la variance est

ωdψd(a), et la covariance vérie :

limn→∞

Cov(F0,n(a), F0,n(b)) =

ωd (1− P(τa = τb =∞))1− P(τ ′0 = τb−a =∞)

P(τ ′0 = τb−a =∞)G(a, b)ωd −G(0, a)G(0, b),

où τz est le temps d'atteinte de z ∈ Zd pour une marche partant de l'origine, et τ ′0le temps de son premier retour à l'origine.

Remarque : Le théorème ci-dessus caractérise le comportement des uctuationsaléatoires de l'odomètre. Il convient de remarquer que ce terme n'est pas nécessai-rement le terme de plus grand ordre après l'équivalent donné au théorème 2.

A une distance plus grande de l'origine, les uctuations ont toujours un com-portement gaussien, avec des variances et covariances diérentes. C'est l'objet duthéorème qui suit.

Théorème . 5 Soit (yn)n∈N une suite de Zd et α > 0 tels que yn → ∞ et yn =

o(n1−α). Lorsque n tend vers l'inni, les uctuations de l'odomètre au voisinage de

yn sont gaussiennes. Plus précisément, pour a ∈ Zd, on dénit la variable aléatoire :

Fyn,n(a) =1

n

(||yn||n

)d/2−1(un

(yn + a

n

)− E

(un

(yn + a

n

)))Alors Fyn,n converge en loi vers un champ gaussien sur Zd, dont la variance est

2d−2 (3G(0, 0)− 1) et la covariance vérie :

limn→∞

Cov(Fyn(a), Fyn(b)) =

4

d− 2

1

1 + P(τb−a <∞)

1− P(τ ′0 = τb−a =∞)

P(τ ′0 = τb−a =∞)G(a, b)

où τz est le temps d'atteinte de z ∈ Zd pour une marche partant de l'origine, et τ ′0le temps de son premier retour à l'origine.

Les preuves des théorèmes 4 et 5 reposent elles aussi sur un encadrement deun entre u+

n et u−n . Les uctuations de u−n , par exemple, sont identiables à l'aide


du théorème de Lindeberg-Feller, puisque cette variable aléatoire est une somme devariables aléatoires indépendantes. La question qui se pose alors est de savoir si unest susamment proche de u−n pour avoir les mêmes uctuations. Pour ce faire, nousétudions la diérence entre le nombre de passages d'une marche en un point z avantsa sortie de Bk−ln(k)2 d'une part, et le nombre de passage de cette même marche enz avant sa sortie de Bk+ln(k) d'autre part. Ainsi, la diérence entre u−n et u+

n peutêtre majorée par une somme de variables aléatoires de la forme qui suit.

χjk(z) =

ξjk+1+(ln k)2∑t=ξj

k−(ln k)2

1Sj(t)=z

Une rapide étude de l'espérance de ces variables aléatoires permet de conclure quece terme d'erreur est négligeable, quand d > 3 et dans le cas de points z à distancenon macroscopique de l'origine.

1.4.3 La forme limite de l'iDLA non centré : une vraie boule dechaleur

Dans ce chapitre, nous présentons un théorème de convergence pour le modèle del'iDLA construit avec des marches aléatoires non centrées. La forme limite est carac-térisée comme une vraie boule de chaleur, car elle vérie une propriété de la moyennepour les fonctions qui vérient l'équation de la chaleur (fonctions caloriques). L'exis-tence d'une telle forme, qui soit de plus bornée, est un problème ouvert de théoriedes EDP paraboliques (voir [8]), auquel notre forme limite apporte une solution.

Le principal résultat de ce chapitre est le suivant :

Théorème . Soit An l'agrégat normalisé d'iDLA, construit à l'aide de marches aléa-

toires non centrées. Alors il existe un ensemble D ⊂ Rd−1 ×R+ qui a les propriétés

suivantes :

1. Presque sûrement, An converge vers D pour la distance de Hausdor.

2. Si φ est une fonctions C∞ de l'espace et du temps qui vérie l'équation dié-

rentielle1− p

2(d− 1)∆φ+ p

∂φ

∂t= 0,

Alors φ vérie la propriété de la valeur moyenne suivante :∫Dφ(z, t)d(z, t) = |D|φ(0).

3. L'ensemble D est borné dans les dimensions de temps et d'espace.

Le premier point du théorème présente notre résultat de convergence pour l'agré-gat normalisé. Le modèle admet donc une forme limite dont les propriétés sont pré-cisées dans les points 2 et 3. Le second point présente une propriété de la moyenne


pour les fonctions caloriques à l'aide de la forme D, ce qui justie son appellationde vraie boule de chaleur. Le dernier point répond au problème ouvert de l'existenced'une telle propriété de la moyenne sur un ensemble borné. En résumé, l'agrégatnormalisé converge presque sûrement vers une vraie boule de chaleur bornée. Nousdonnons une simulation de cette forme en dimension 2 dans la gure 1.5.

Figure 1.5 Agrégat de percolation pour 500 000 particules, p = 0.2

Notre preuve suit l'idée générale de Levine et Peres [17], dont la méthode pouridentier les formes limites peut être modiée pour s'appliquer à notre contexte.Tout d'abord, nous introduisons un modèle appelé unfair divisible sandpile, c'est àdire modèle du tas de sable divisible partial, parce qu'il privilégie une direction parrapport aux autres.

Pour dénir ce modèle, on considère une distribution de masse sur Zd, dont lamasse totale est nie et qui a un support borné. Un site est dit plein s'il a unemasse au moins 1. Un site plein peu s'écrouler, c'est à dire garder une masse 1

et redistribuer son excès de masse entre ses voisins, selon la proportion suivante :p dans la direction privilégiée (dénotée par le vecteur directeur ed), et

1−p2(d−1) vers

chacun des ses voisins sur l'hyperplan perpendiculaire à la direction privilégiée.

A chaque étape, on réalise l'écroulement d'un site plein. Lorsque le nombred'étapes tend vers l'inni, à condition que chaque site plein soit visité une in-nité de fois, on prouve que la répartition de masse converge vers une distributionasymptotique dans laquelle chaque site a une masse inférieure ou égale à 1.

Comme le modèle classique du tas de sable divisible, notre modèle est abélien,c'est à dire que la conguration nale est unique et ne dépend pas de l'ordre des


écroulements.

Un outil crucial pour l'étude de ce modèle est la fonction odomètre. On la dénitune fois de plus comme la masse émise par un point x pendant les écroulements quiamènent à la répartition nale.

u(x) = masse totale émise par x.

Comme on sait comment est répartie la masse qui sort de x et de chacun de sesvoisins, on connaît la masse totale reçue par x, c'est à dire :

masse reçue en x = pu(x− ed) +1− p

2(d− 1)

∑y x,y−x⊥ed

u(y).

On obtient alors le bilan de masse suivant :

ν(x)− σ(x) = −p (u(x)− u(x− ed)) + (1− p)∆u(x)

avec σ et ν les masses initiales et nales en x, et ∆ le Laplacien dis-cret sur les d − 1 coordonnées perpendiculaires à ed, c'est à dire :∆u(x) =

12(d−1)

∑y x,y−x⊥ed (u(y)− u(x)).

On dénit donc un opérateur discret K qui résume cette opération :

Kf(x) = (1− p)∆f(x)− p (f(x)− f(x− e1)) .

Étant donné la nature de K, on est amené à considérer la dernière coordon-née comme une coordonnée de temps, et les d − 1 autres coordonnées comme lescoordonnées d'espace. Alors le pendant continu de notre opérateur discret sera :

Kf(x, t) =1− p

2(d− 1)∆f(x, t)− p∂f

∂t(x, t),

qui est plus connu sous le nom d'opérateur de la chaleur.

On prouve ensuite que la répartition de masse dénie par le modèle du tas desable partial est donnée par la solution d'un problème d'obstacle parabolique discret.Cette solution converge, lorsque la taille du réseau tend vers 0, vers la solution d'unproblème d'obstacle parabolique continu.

L'étape suivante de notre preuve consiste à relier le modèle de l'iDLA non centréet le modèle du tas de sable divisible partial, ce qui nous permet de prouver que cesdeux modèles ont la même limite d'échelle D.

Une vision heuristique de ce lien entre les modèles peut être donnée de la manièresuivante, en dénissant u(x) comme le nombre de marches qui passent par x lors dela construction de l'agrégat, sans s'y arrêter. Une marche aléatoire passant au pointx répartit son prochain pas selon une loi qui correspond à la répartition de massedécrite plus haut, c'est à dire p dans la direction privilégiée, et 1−p

2(d−1) dans chacunedes autres directions. Si l'on suppose qu'un grand nombre de marches est passé en


x, et qu'il en va de même pour les voisins de x, on devrait avoir en un certain sens :

Ku(x)?= ν(x)− σ(x),

où K, ν et σ sont les quantités décrites plus haut. Cette équation n'est pas correctepuisqu'elle présupposerait l'indépendance de quantités qui sont en fait dépendantes,mais elle donne une idée de la façon dont les modèles sont reliés.

Pour arriver à une preuve rigoureuse du résultat, des arguments classiques del'iDLA sont utilisés. Il s'agit principalement de l'utilisation de la construction al-ternative en arrêtant les marches quand elles sortent d'un ensemble, et des troisvariables aléatoires L,M,N décrites dans la section 1.2.1.

On conclut notre étude en utilisant des estimées sur les marches aléatoires, etles méthodes développées par Lawler dans la seconde partie de [15], an de prouverd'un point de vue probabiliste que cette forme limite D est nécessairement bornéedans les directions de temps et d'espace.

1.4.4 A la recherche d'une borne supérieure pour l'iDLA

Dans ce chapitre, qui est le fruit d'une collaboration avec Hugo Duminil, ArielYadin et Amir Yehudayo, nous nous penchons sur le problème de la convergencevers une forme limite du point de vue de la borne extérieure de l'iDLA.

Plus particulièrement, nous nous plaçons dans le cas où l'on connaît une borneintérieure ; on a alors besoin d'une borne extérieure pour pouvoir établir un résultatde forme limite.

Remarquons que, si l'étude de la borne extérieure sur Zd n'est pas en elle mêmeplus dicile que celle de la borne inférieure, elles utilisent des ingrédients diérents.L'étude de la borne inférieure repose généralement sur la connaissance des fonctionsde Green et des fonctions harmoniques, tandis que celle de la borne extérieure né-cessite l'utilisation de bornes sur les mesures harmoniques. Alors que la fonctionde Green est relativement bien comprise dans certains environnements aléatoires,et en particulier dans le cas de la percolation surcritique, ce n'est pas le cas desmesures harmoniques. Par exemple, on ne dispose pas de borne susante sur lamesure harmonique de la boule euclidienne de centre 0 et de rayon n dans l'amasinni de percolation surcritique pour appliquer le raisonnement classique décrit parexemple dans [15]. Nous utilisons donc de nouvelles techniques qui nous permettentde contourner cette diculté.

Notre principal résultat met en relation l'existence d'une borne extérieure aveccelle d'une borne intérieure. Il s'agit en fait de montrer que si l'agrégat contientdes boules (pour une certaine métrique), et que ces boulent représentent presquetoute la masse de l'agrégat (il reste peu de particules à traiter), alors on peu déduirel'existence d'une forme limite.

An d'énoncer notre résultat, nous présentons les conditions nécessaires à sonapplication.

Soit ρ une métrique sur le graphe. Les boules de ρ seront les candidates au titre de


forme limite ; on suppose qu'elles sont contenues dans l'agrégat asymptotiquement.en fait, nous aurons besoin d'une condition un peu plus forte, décrite ci-dessous.Mais d'abord, donnons deux conditions sur la régularité de ρ.(i) Continuité : la métrique est dominée par la distance de graphe. (ii)Croissance du volume : le volume d'une boule de ρ croît comme une puissanced de son rayon ; c'est-à-dire qu'il existe une constance c > telle que

1

crd 6 |Bρ(x, r)| 6 crd ∀ x ∈ Bρ(0, n) et n1/d3 ≤ r ≤ n.

Donnons maintenant les conditions de convergence intérieure de l'agrégat. Pourles exprimer, on dénit l'agrégat arrêté sur le bord d'une boule, construit en arrêtantles particules lorsqu'elles rencontrent le bord d'une boule donnée (voir la dénitiondonnée dans la section 1.2.1). Dans cette partie, on note l'agrégat A, obtenu enlançant k particules depuis le point x, et arrêté au bord de Bρ(x, r) de la façonsuivante :

Ak(x 7→ r).

Nos conditions portent sur cette version modiée de l'agrégat. On note bx(r) =

|Bρ(x, r)|.(iii) Convergence intérieure faible uniforme : il existe α > 0 tel que pour toutn,

Pr[Bρ(x, r) ⊂ Abx(r/α)(x 7→ r)] ≥ α ∀ x ∈ Bρ(0, n+ r) et n1/d36 r 6 n.

Cette condition stipule qu'une version faible de la convergence intérieure est vraiepour tous les points situés à distance au plus 2n de l'origine. En d'autre termes,avec probabilité strictement positive, lancer un nombre de particules de l'ordre debx(r) au point x donnera naissance à un agrégat arrêté contenant la boule Bρ(x, r).(iv) Convergence intérieure forte locale : l'agrégat arrêté à la boule Bρ(0, n)

contient une proportion asymptotiquement proche de 1 des points de la boule. End'autres termes, presque sûrement,

|Abo(n)(o 7→ n)|/bo(n)→ 1.

Cette condition est un peu plus forte que la simple convergence intérieure, maisdans la pratique la plupart des preuves de la convergence intérieure sont en fait despreuves de cette condition (c'est vrai par exemple dans [15], [17], [20]...).

Notre résultat s'exprime alors de la façon suivante :

Théorème . Sous les conditions (i) à (iv), on a une borne extérieure précise sur

l'agrégat d'iDLA. Plus précisément, pour tout ε > 0, on a presque sûrement, à partir

d'un certain rang :

A(n) ⊂ Bρ(0, (1 + ε)n).

Une rapide vérication montre que les conditions (i) à (iv) sont en particuliervériées pour le modèle de l'iDLA sur l'amas de percolation surcritique, ce qui


permet d'obtenir le pendant extérieur du résultat intérieur de Shellef. La gure 1.6est une simulation de l'iDLA sur l'amas de percolation.

Figure 1.6 Agrégat d'iDLA à 6 300 particules sur l'amas de percolation surcritique

(p=0.6). Les points rouges sont les points de l'agrégat, les points verts ceux de l'amas

de percolation.

La preuve de ce théorème repose sur le raisonnement suivant : lorsque l'onconstruit l'agrégat, on peut dans un premier temps arrêter les particules à la sortiede Bρ(0, n). Pour obtenir l'agrégat nal, il faut alors relancer ces particules en lesautorisant cette fois à sortir de Bρ(0, n).

Remarquons tout d'abord que ce nombre de particules restantes peut être pris in-férieur à η0n. En eet, la borne inférieure garantit qu'une proportion arbitrairementproche de 1 de particules contribue à l'agrégat avant de sortir de Bρ(0, n).

On procède à la construction de l'agrégat nal par étapes. A chaque étape k, onlance les particules restantes et on les laisse évoluer jusqu'à leur sortie de Rρ(0, n+

εk(n).On montre qu'alors, avec grande probabilité, une proportion α > 0 constante à

chaque étape de particules contribuent à l'agrégat avant de sortir de Rρ(0, n+εk(n).Une étude détaillée montre que pour tout η > 0, il existe une suite εk(n) vériantcette propriété et telle que de plus,

∑k>0 εk(n) 6 ηn. On procède ainsi tant qu'il

reste plus de ηn1/d particules à traiter, c'est à dire les particules qui n'ont pas encoreajouté à l'agrégat. Lorsque le nombre de particules devient plus petit, on sait queles particules restantes ne peuvent pas atteindre la distance ηn de leur point de

Bibliographie 25

départ (c'est une conséquence simple de la règle d'agrégation). Ainsi, l'agrégat seraasymptotiquement contenu dans Bρ(0, n+2ηn), où η peut être choisi arbitrairementpetit.

Bibliographie

[1] A. Asselah and A. Gaudilliere. From logarithmic to subdiusive polyno-mial uctuations for internal dla and related growth models. Arxiv preprint

arXiv :1009.2838, 2010.

[2] A. Asselah and A. Gaudilliere. Sub-logarithmic uctuations for internal dla.Arxiv preprint arXiv :1011.4592, 2010.

[3] S. Blachere. Agrégation limitée par diusion interne sur zd. In Annales de

l'Institut Henri Poincare (B) Probability and Statistics, volume 38, pages 613648. Elsevier, 2002.

[4] A. A. Chernov. Reduplication of multicomponent chains by the lighteningmechanism]. Biozika, 12(2) :297, 1967.

[5] P.G. de Gennes. La percolation : un concept unicateur. La Recherche, 7(72) :919927, 1976.

[6] P. Diaconis and W. Fulton. A growth model, a game, an algebra, lagrangeinversion, and characteristic classes. Rend. Sem. Mat. Univ. Pol. Torino, 49(1) :95119, 1991.

[7] G. Grimmett. Percolation, volume 321. Springer Verlag, 1999.

[8] A. Hakobyan and H. Shahgholian. Generalized mean value property for caloricfunctions. Survey), submitted.

[9] D. Jerison, L. Levine, and S. Sheeld. Internal dla in higher dimensions. Arxivpreprint arXiv :1012.3453, 2010.

[10] D. Jerison, L. Levine, and S. Sheeld. Logarithmic uctuations for internaldla. Arxiv preprint arXiv :1010.2483, 2010.

[11] D. Jerison, L. Levine, and S. Sheeld. Internal dla and the gaussian free eld.Arxiv preprint arXiv :1101.0596, 2011.

[12] D. Landolt. Fundamental aspects of electropolishing. Electrochimica Acta, 32(1) :111, 1987.

[13] G.F. Lawler. Subdiusive uctuations for internal diusion limited aggregation.The Annals of Probability, pages 7186, 1995.

[14] G.F. Lawler. Intersections of random walks. Birkhauser, 1996.

26 Bibliographie

[15] G.F. Lawler, M. Bramson, and D. Grieath. Internal diusion limited aggre-gation. The Annals of Probability, pages 21172140, 1992.

[16] L. Levine and Y. Peres. Strong spherical asymptotics for rotor-router aggrega-tion and the divisible sandpile. Potential Analysis, 30(1) :127, 2009.

[17] L. Levine and Y. Peres. Scaling limits for internal aggregation models withmultiple sources. Journal d'Analyse Mathématique, 111(1) :151219, 2010.

[18] P. Meakin and J. M. Deutch. The formation of surfaces by diusion-limitedannihilation. J. Chem. Phys., 85, 1986.

[19] V.B. Priezzhev, D. Dhar, A. Dhar, and S. Krishnamurthy. Eulerian walkers asa model of self-organized criticality. Physical review letters, 77(25) :50795082,1996.

[20] E. Shellef. Idla on the supercritical percolation cluster. Electronic Journal of

Probability, 15 :723740, 2010.

[21] Y.G. Sinai. Limit behaviour of one-dimensional random walks in random envi-ronments. Teoriya Veroyatnostei i ee Primeneniya, 27(2) :247258, 1982.

[22] F. Solomon. Random walks in a random environment. The annals of probability,pages 131, 1975.

Chapitre 2

The Arcsine law as the limit ofthe internal DLA clustergenerated by Sinai's walk

We identify the limit of the internal DLA cluster generated by Sinai's walk as thelaw of a functional of a Brownian motion which turns out to be a new interpretationof the Arcsine law.

Contents2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Notations and main results . . . . . . . . . . . . . . . . . . . 28

2.3 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.1 Good environments . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.2 The quenched localization of dn . . . . . . . . . . . . . . . . . 33

2.3.3 Characterization of the law of d∗ . . . . . . . . . . . . . . . . 35

2.4 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . 37

Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1 Introduction

The internal diusion limited aggregation method was rst introduced by Dia-conis and Fulton in 1982 (see [2]) and gives a protocol for building a sequence ofgrowing random sets A(n), n > 0 using random walks. At each iteration, the setA(n+1) is obtained from A(n) by addition of the rst site visited by a walk startingfrom 0 outside A(n). When the walk is a simple random walk on Zd, the cluster hasthe ball as asymptotic limit shape ([3]). In the special and trivial case of dimensionone, the cluster A(n) is an interval denoted by [gn, dn] and dn/n converges almostsurely to 1/2. In this paper, we consider the case where the cluster is generatedby one dimensional walks evolving in an inhomogeneous random medium. Moreprecisely we are dealing with one dimensional recurrent random walks in randomenvironment often called Sinai's walk. One can rapidly understand that dependingon the prole of the potential associated to the environment the cluster may be dras-tically asymmetric. One can even prove that lim sup dn/n = 1 and lim inf dn/n = 0

almost surely (Theorem 2.2). But beyond this rough result we are able to prove,

28 Chapitre 2. iDLA in Sinai's environment

under the annealed measure, a convergence in law for dn/n towards the Arcsine law(Theorem 2.1). The Arcsine law comes out from a new functional of the Brownianmotion which in our case is the scaling limit of the potential. This functional stemsfrom the exploration of the potential by the growing cluster. It involves in Sinai'sterminology the largest valley of width smaller than one containing the origin. Letus recall that the limit law of the walk in Sinai's theorem after scaling by ln2 n

involves the smallest valley of height one.Finally, we mention that this result provides a new understanding of the Arcsine

law which is dierent from the two classical ones (last zero of the Brownian motionbefore 1, and the time spent by the Brownian motion above 0 before 1).

2.2 Notations and main results

We rst introduce Sinai's random walk in random environment. An environmentω is a collection (ω(i))i∈Z of numbers in [0, 1]. We denote by Ω := [0, 1]Z the set ofenvironments.

For a given environment ω, we dene a Markov chain (Xn)n∈N of law Pω, oftencalled quenched law, by

Pω(X0 = 0) = 1,

and for every x ∈ Z and n ∈ N,

Pω(Xn+1 = x+ 1|Xn = x) = ω(x),

Pω(Xn+1 = x− 1|Xn = x) = 1− ω(x).

We endow Ω with its canonical σ−eld and a probability measure P of the formP := µ⊗Z, where µ is a probability on [0, 1]. We can now dene a probability P onthe space of trajectories, called the annealed law, by :

P =

∫ΩPωdP.

Introducing the notation ρ(i) := 1−ω(i)ω(i) , we make the following assumptions on

µ :

(i) µ(ω(0) = 0) = µ(ω(0) = 1) = 0

(ii) Eµ(log ρ(0)) = 0

(iii) Eµ[(log ρ(0))2

]<∞

A random walk in random environment satisfying the assumptions above isusually called a Sinai walk, referring to the famous article of Sinai [6] proving theconvergence in law of Xn/(log n)2 under the annealed law.

Assumption of ellipticity (i) is an irreducibility assumption.

2.2. Notations and main results 29

Assumption (ii) ensures that Pω is recurrent P-almost surely (see [7] for asurvey on one dimensional random walks in random environments).

Assumption (iii) makes it possible to apply Donsker's principle to the potentialVω (see (2.2.1) below).

Let us now explain the construction of the internal diusion limited aggregationcluster for a given environment ω ∈ [0, 1]Z.

Let (Xj(n))n∈N be an i.i.d. family of random walks such that ∀j ∈ N, (Xj)

has law Pω. We now dene our cluster A(n) as a classical internal diusion limitedaggregation cluster using this family of random walks. Dene A(n) and the stoppingtimes (θk)k∈N recursively in the following way :

θ0 = 0,

A(0) = 0 = X0(θ0), and for all j > 0,

θj = infn > 0 : Xj(n) 6∈ A(j − 1),A(j) = A(j − 1) ∪ Xj(θj).

We stress the fact that under the annealed law, the same environment is used throu-ghout the construction of A(n).

As the construction of the cluster uses an i.i.d sequence of walks, the followingnotations will be helpful to state our theorems,

Pω = P⊗Nω and P =

∫ΩPωdP.

The crucial tool introduced by Sinai in [6] is the potential associated to a givenenvironment :

Vω(0) = 0

Vω(i) =

i∑k=1

ln ρ(k) if i > 1, (2.2.1)

Vω(i) = −0∑

k=i+1

ln ρ(k) if i 6 − 1.

For all n > 0, we dene the renormalized potential

V (n)ω (t) =

1√nVω(bntc).

It follows from Donsker's principle and Assumption (iii) that, as n goes to in-

nity, (V(n)ω (t))t∈R converges in law to the Wiener law. Let us notice nally that, by

construction, the cluster A(n) is an interval we will denote by [gn, dn].


We can now state our main result :

Theorem 2.1. Under P , dn/n converges in law to the Arcsine law. Namely, for all

0 6 a 6 b 6 1,

P

(dnn∈ (a, b)

)−−−→n→∞

∫ b

a

1

π√x(1− x)

dx.

Moreover the following theorem describes the almost sure behavior of the DLAcluster in a typical environment :

Theorem 2.2. P−a.s., with Pω probability one,

lim supn→∞

dnn

= 1, and

lim infn→∞

dnn

= 0.

2.3 Proof of Theorem 2.1

The proof of Theorem 2.1 can be decomposed in the three following steps.

2.3.1 Good environments

For each n, we dene the set of good environments which will turn out to be ofhigh probability and on which we will be able to control the position of the clusterat step n.

For all cadlag functions v : R→ R, we dene,

T+y (v) = inf t > 0, such that v(t) > y ,T−y (v) = − sup t 6 0, such that v(t) > y ,

y = supy > 0, such that T+y + T−y 6 1. (2.3.1)

We introduce the length of the excursion of v below its maximum at T+y :

α = inft > 0, such that v(T+

y + t) > v(T+y ),

and the length of the excursion of v below its maximum to the left of −T−y :

β = inft > 0, such that v(T−y − t) > v(T−y )

.

In order to make the computation more readable, we will use for all n ∈ N thefollowing notations :

T+yn = T+

y (V (n)), T−yn = T−y (V (n)), yn = y(V (n)),

αn = α(V (n)), βn = β(V (n)).

2.3. Proof of Theorem 2.1 31

Let (Bt)t∈R be a real standard Brownian motion dened on an abstract probabilityspace (Ω,F,P). We introduce the notations,

T+y = T+

y (B), T−y = T−y (B), y = y(B),

α = α(B), β = β(B)

These notations (as well as d∗ and g∗ introduced in the next section) are illustratedin Figure 2.1 in the case of the Brownian motion.

Bt

1

t0

y

T−y = g∗ T+

y d∗

α

Figure 2.1 On this example α > 0 while β = 0. The bold part of the path

corresponds to the theoretical part of the potential that is explored by the cluster.

Remark 1. Throughout the paper we make use of Donsker's principle for the func-

tionals α,β,T+y ,T

−y , y, as well as d∗ (introduced in the next section), which are not

continuous with respect to the Skorohod topology. However, we observe, for all these

functionals, that the set on which they are discontinuous is a subset of trajectories

having two local maxima at the same height which has Wiener measure 0. Hence,

Donsker's principle applies to these functionals.


For all ε > 0 and n > 0, we dene the following events :

Bε,+n =

sup[−T−yn−βn−ε,−T

−yn−βn]

V (n)ω > yn +

n1/3

√n

∩ βn < ε ,

Bε,−n =

sup[T+yn

+αn,T+yn

+αn+ε]V (n)ω > yn +

n1/3

√n

∩ αn < ε ,

Cε,+n =

sup[−T−yn+ε,0]

V (n)ω 6 yn −

n1/3

√n

,

Cε,−n =

sup[0,T+

yn−ε]

V (n)ω 6 yn −

n1/3

√n

.

These events can be described as follows. On Bε,+n , the length of the excursion below

the supremum to the left of −T−yn is smaller than ε. Furthermore, to the left of this

excursion the potential increases enough to build an obstacle for the walk. On Cε,+n ,there is no signicant obstacle between 0 and −T−yn + ε. On the intersection of thesetwo events, we expect the left border of the cluster to be close to −T−yn while the rightborder should go beyond T+

yn . The events indexed by “− refer to the symmetricsituation and can be described in the same way.

Lemma 2.3. For all ε > 0,

limn→∞

P((Bε,+n ∩ Cε,+n

)∪(Bε,−n ∩ Cε,−n

))= 1

Proof of Lemma 2.3. We dene the following ltration :

F0 = σ (Vω(i), i > 0) (2.3.2)

Fk = σ (Vω(i), i > − k) for all k > 0. (2.3.3)

The process (Vω(−i))i > 0 is Markovian and adapted to F := (Fj)j > 0. The

Markov property at the F-stopping time (−nT−yn − nβn) and classical properties ofrandom walks yield

P

sup[−T−yn−βn−ε,−T

−yn−βn]

V (n)ω > yn + n1/3/

√n

−−−→n→∞

1.

The same argument holds for the symmetric case.It follows from Donsker's principle that

P (βn < ε ∪ αn < ε)→ P (β < ε ∪ α < ε) .

As(T+y (B)

)y > 0

and(T−y (B)

)y > 0

are two strictly increasing subordinators wi-


thout drift, their sum is also a strictly increasing subordinator without drift, andgoes almost surely above the level 1 while jumping (see Proposition 1.9 in [1]).Furthermore, they are independent so they never jump at the same time (for back-ground on subordinators, we refer to [1]). As a consequence, almost surely underWiener measure, either α > 0 and β = 0 or α = 0 and β > 0, hence

P (β < ε ∪ α < ε)→ 1, as ε tends to 0.

It follows from the same argument that, a.s., one of T+y and T−y is a local

maximum (while the other is not). Notice also that(−n1/3/

√n,+∞

)increases to

(0,+∞), hence from Donsker's principle,

P(Cε,−n ∩ Cε,+n

)−−−→n→∞

P

sup[−T−y +ε,T+

y−ε]B 6 y

.

This happens almost surely because the Brownian motion already has a local maxi-mum with value y at T+

y or T−y .

2.3.2 The quenched localization of dn

We can now limit our study to the good environments, where with high probabi-lity dn is localized near its theoretical position d∗n that is a deterministic functionalof the potential. More precisely, we dene for any cadlag function v from R to R,

d∗(v) = T+y (v) + 1α(v)>β(v)(1− (T+

y (v) + T−y (v))),

and we will use the following notations :

d∗n = d∗(V (n)), d∗ = d∗(B), g∗ = 1− d∗.

We will also use the notation

g∗n = d∗n − 1.

Proposition 2.4. For all ε > 0 and η > 0,

P(Pω

(∣∣∣∣dnn − d∗n∣∣∣∣ > ε

)> η

)−−−→n→∞

0.

Proof of Proposition 2.4. If (Xn)n > 0 is a Markov chain on Z, we will use, for anyq in Z, the notation σq to denote the hitting time of q by Xn, namely

σq = infn > 0, Xn = q.


Using the fact that the function hω dened on Z by

hω(k) =k−1∑i=0

exp(Vω(i)) for k > 0

hω(0) = 0

hω(k) = −−1∑i=k

exp(Vω(i)) for k < 0

is harmonic for Xn under Pω (i.e. hω(Xn) is a martingale under Pω, which is aclassical fact in birth and death processes, we deduce from the optional samplingtheorem that, for any ω ∈ Ω,

Pω(σ−b < σa) =

∑a−1i=0 exp(Vω(i))∑a−1i=−b exp(Vω(i))

(2.3.4)

(see for example [7] p. 196, for a use of this identity in the framework of randomwalks in random environment).

Let n be in N \ 0 and ω be in Bε,+n ∩ Cε,+n , then

Pω(gn < −nT−yn − 2εn) 6 n2 exp(−n1/3). (2.3.5)

Indeed suppose that Vω(T+yn) = yn, then

Pω(σ−nT−yn−2εn < σn∧(nT+yn

+nαn−1)) =

∑n∧(nT+yn

+nαn−1)−1

i=0 exp(Vω(i))∑n∧(nT+yn

+nαn−1)−1

i=−nT−yn−2εnexp(Vω(i))

6n exp(yn)

exp(yn + n1/3)= n exp(−n1/3)

Now as n(T+yn + T−yn + 2ε+ αn) > n, the probability in (2.3.5) can be controlled by

the probability that one of n independent random walks in the environment ω exitsof the interval [−nT−yn − 2εn, nT+

yn + nαn − 1] by the left side, namely

Pω(gn < −nT−yn − 2εn) < 1− (1− n exp(−n1/3))n 6 n2 exp(−n1/3) (2.3.6)

Suppose now that Vω(T+yn) > yn then

Pω(σ−nT−yn−2εn < σnT+yn−1) =

∑nT+yn−2

i=0 exp(Vω(i))∑nT+yn−2

i=−nT−yn−2εnexp(Vω(i))

6n exp(yn)

exp(yn + n1/3)= n exp(−n1/3)


It follows from the denition of yn that n(T+yn + T−yn + βn) > n which leads to

the same kind of control as in (2.3.6) and concludes the proof of (2.3.5). We nowcomplete the study of gn on Bε,+

n ∩ Cε,+n by proving the following inequality,

Pω(gn > nT−yn + εn) 6 n2 exp(−n1/3). (2.3.7)

Using (2.3.4) again, we get

Pω(σnT+yn< σ−nT−yn+εn) =

∑−nT−yn+εn−1

i=0 exp(Vω(i))∑−nT−yn+εn−1

i=nT+yn

exp(Vω(i))

6n exp(yn − n1/3)

exp(yn)= n exp(−n1/3).

As nT+yn +nT−yn + εn > n, we control the probability of the complementary event in

(2.3.7) with the probability that n independent random walks in the environmentω exit [−nT−yn + εn, nT+

yn ] through the left side, namely

Pω(gn > − nT−yn + εn) 6 1− (1− n exp(−n1/3))n 6 n2 exp(−n1/3).

Notice now that on Bε,+n ∩Cε,+n , |T+

yn−g∗n| < ε. With a similar study of the event

Bε,−n ∩Cε,−n and Lemma 2.3, it is easy to complete the proof of Proposition 2.4.

2.3.3 Characterization of the law of d∗

Using Donsker's principle and Proposition 2.4, we conclude that dn/n convergesin law towards d∗. To complete the proof of Theorem 2.1, we characterize the lawof d∗.

Lemma 2.5. The law of d∗ is the Arcsine law.

Proof of Lemma 2.5. We note (Lt)t > 0 the local time at 0 of B and

τt = supu > 0, Lu < t

the left continuous inverse of L.

Remark 2. The process (T+y (B))y > 0 is left continuous. Consequently, even if it

is unusual, we prefer to work with the left continuous version of the inverse local

time of Bt. We will also use the name subordinator for a left continuous increasing

process with independent increments.

For all t > 0, denote

A+t =

∫ t

01Bs>0ds,


the time spent by B above 0 up to time t ; and similarly

A−t =

∫ t

01Bs<0ds.

Now A+τt +A−τt = τt is a subordinator and, as τt follows the same law as the hitting

time of t by the Brownian motion, it is a 1/2−stable subordinator. Moreover it is aconsequence of Ito's excursion theorem (see Theorem 2.4 of [5]) that both processes(A+

τt) and (A−τt) are independent subordinators with the same law. Each of them dealsindeed respectively with one of the disjoint set of positive and negative excursionsof B up to time τt. Hence

(A+τt , A

−τt)

(law)=

1

4(T+t (B), T−t (B)) (2.3.8)

(we refer to the lecture notes of Marc Yor [8] for explanations and applications ofthis identity). Hence

y(law)= supt > 0, A+

τt +A−τt 61

4

(law)= supt > 0, τt 6

1

4

(law)= L1/4.

From the choice of the left continuous version of the inverse local time (see Remark2) we deduce

T+y

(law)= 4A+

τL1/4

(law)= A+

g1(2.3.9)

whereg1 = supt < 1, Bt = 0

is the last zero of (Bt)t > 0 before 1. Let us remind that

d∗ = T+y + 1α>β(1− (T+

y + T−y )). (2.3.10)

Notice now that (T+y ,T

−y ) is independent of 1α>β. Indeed, using (2.3.8), it is

equivalent to check the independence of the sign of B1 and (A+g1, A−g1

). Gatheringthis independence with (2.3.10) and (2.3.9), we obtain

d∗(law)= A+

g1+ χ(1− (A+

g1+A−g1

)) = A+g1

+ χ(1− g1), (2.3.11)

where χ is a Bernoulli variable with parameter 1/2 independent of all other variables.The decomposition of the path of the Brownian motion on [0, 1] into a path on [0, g1]

and an incomplete excursion of independent sign on [g1, 1] yields

A+g1

+ χ(1− g1)(law)= A+

1 . (2.3.12)


We conclude by recalling Paul Lévy's well known result (formula (53) p. 325 of [4],which states that A+

1 follows the Arcsine law.

Remark 3. Notice that the Proof of Lemma 2.5 has a non-trajectorial nature as

shown by the key identity in law (2.3.8), which allows to reduce the study of the

functional of two independent Brownian motions to that of the functional of a single

one.

Remark 4. The random sign χ in identity (2.3.11) corresponds to the left-right

symmetry of the problem, whereas in the classical decomposition (2.3.12) it has an

up-down meaning.


We will only prove the rst statement as the second one can be easily deducedusing the symmetry of the model.

Lemma 2.6. P-a.s.,T+

yn > 1− ε i.o.

Proof of Lemma 2.6. Let k be in N∗. From Donsker's invariance principle, we deducethat there exist k independent Brownian motions (B1, · · · , Bk) on (Ω,F,P) suchthat

(V (n), · · · , V (nk))(law)⇒ (B1, · · · , Bk).

We will also use the fact that for any real Brownian motion B,

P(T+y (B) > 1− ε) > 0,

see for example (2.3.9). Now

lim infT+yn 6 1− ε ⊂ lim inf

T+

yn 6 1− ε ∩ · · · ∩ T+ykn6 1− ε

and

P(lim infT+yn 6 1− ε) 6 lim inf P

(T+yn 6 1− ε

∩ · · · ∩

T+ykn6 1− ε

)6 P

(T+y (B1) 6 1− ε

)· · ·P

(T+y (Bk) 6 1− ε

)6 P

(T+y (B1) 6 1− ε

)kAs k can be chosen arbitrarily big, this concludes the proof of Lemma 2.6.

We deduce from Lemma 2.6 and the previous study of Cε,−n (see the Proof ofTheorem 2.1) that Cε,−n ∩ T+

yn > 1− ε occurs innitely often. Fix ω in Ω and n in

38 Bibliographie

N such that ω ∈ Cε,−n ∩ T+yn > 1− ε. Formula (2.3.4) yields

Pω(σnT−yn< σn(T+

yn−ε)) 6

∑n(Tyn−ε)−1i=0 exp(Vω(i))∑n(Tyn−ε)−1

i=nT−ynexp(Vω(i))

6neyn−n

1/3

eyn6 ne−n

1/3

The probability of the event dn 6 T+yn − ε is smaller than the probability that

one of n independent random walks in the environment ω exits [−nT+yn , n(T+

yn − ε)]through the left side. Hence,

Pω(dn 6 T+yn − ε) 6 1− (1− ne−n1/3

)n

6 n2e−n1/3.

We conclude using Borel Cantelli's Lemma on a subsequence (nj)j > 0 such thatCε,+nj ∩ T+

yn > 1− ε holds for all j > 0 (the P-a.s. existence of such a subsequenceis a consequence of Lemma 2.6).

Acknowledgements

It is a pleasure to thank Professor Marc Yor for pointing out to us the keyidentity (2.3.8).

Bibliographie

[1] J. Bertoin. Subordinators : examples and applications. Lectures on probability

theory and statistics, pages 191, 2004.


[3] G.F. Lawler, M. Bramson, and D. Grieath. Internal diusion limited aggrega-tion. The Annals of Probability, pages 21172140, 1992.

[4] P. Lévy. Sur certains processus stochastiques homogenes. Compositio Math, 7(283-339) :0000919, 1939.

[5] D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume293. Springer Verlag, 1999.

[6] Y.G. Sinai. Limit behaviour of one-dimensional random walks in random envi-ronments. Teoriya Veroyatnostei i ee Primeneniya, 27(2) :247258, 1982.

Bibliographie 39

[7] S. Tavaré and O. Zeitouni. Lectures on probability theory and statistics (paper-back) : Ecole d'eté de probabilités de saint-our xxxi-2001 book.(series : Lecturenotes in mathematics). 2004.

[8] M. Yor. Local times and excursions for brownian motion : a concise introduction.Lecciones en Mathematicas, 1995.

Chapitre 3

Asymptotics and uctuations ofthe odometer function for the

iDLA model

We present precise asymptotics of the odometer function for the internal Dif-fusion Limited Aggregation model. These results provide a better understandingof this function whose importance was demonstrated by Levine and Peres [9]. Dis-crete Gaussian elds arise as the law of uctuations of the odometer around variouspoints. We also derive a dierent proof of a time-scale result by Lawler, Bramsonand Grieath [7].


3.2 Denitions, main results . . . . . . . . . . . . . . . . . . . . . 42

3.3 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . 57

3.6 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . 58

3.7 Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . 61

3.8 Back to a result of Lawler, Bramson and Grieath . . . . . 64

Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.1 Introduction

The internal Diusion Limited Aggregation model was rst introduced by Dia-conis and Fulton in [3] and gives a protocol for building a random set recursively. Ateach step, the rst vertex visited outside the cluster by a simple random walk star-ted at the origin is added to the cluster. The resulting limit shape is the Euclideanball, as proved in 1992 by Lawler, Bramson and Grieath in [7].

More recently, however, Levine and Peres [9], [8] have shown that this modelis related to the rotor-router and divisible sandpile models. In the former, randomwalkers are replaced by Eulerian walkers. In the latter, each vertex can hold 1 unitof mass, and the excess is divided equally among its neighbors when the vertex

42 Chapitre 3. Asymptotics and uctuations of the odometer

topples. Thus an initial mass at the origin becomes a stable shape after a suitableinnite series of topplings.

For each of these three models, one can dene an odometer function, whichwill be the total number of times a walker passes through a given point (countingmultiple passages of the same walker) for the internal DLA and rotor-router model,and the total mass emitted from a given point in the construction of the clusterfor the divisible sandpile model. This function plays a comparable role in all threemodels and turns out to be instrumental in their relation. While the limiting shapeof these models is known to be the Euclidean ball, the behavior of their odometerfunctions in the particular case where all the mass is started at the origin remainsto be studied.

This paper provides a closer look at the odometer function in the case of theinternal DLA model, with an almost sure convergence of the normalised functionsand asymptotics of this function near the origin. The uctuations of the odometerare studied at points within nite, mesoscopic and macroscopic distance from theorigin. They are identied as discrete Gaussian elds, whose covariances are compu-ted. Around points within mesoscopic distance from the origin, the Gaussian eldsinvolved have covariance functions reminiscent of that of the discrete Gaussian FreeField. The almost sure convergence results also provide a new proof of the time scaleof the cluster introduced in [7].

3.2 Denitions, main results

Let (Sj)j∈N a sequence of independent simple random walks on Zd, and let usdene the cluster A(n) and stopping times (σk)k∈N recursively in the following way :

σ0 = 0,

A(0) = 0 = S0(σ0), and for all j > 0,

∀j > 0, σj = inft > 0 : Sj(t) 6∈ A(j − 1),A(j) = A(j − 1) ∪ Sj(σj).

Let ||.|| denote the Euclidean norm on Rd, Br the Euclidean ball of radius r ofZd, and ωd the volume of the unit ball of Rd. We will consider the cluster A(ωdn

d),which has the same volume as Bn. Lawler, Bramson and Grieath proved in [7]that the normalized cluster 1

nA(⌊ωdn

d⌋) converges to the Euclidean unit ball with

probability one.Let z ∈ Rd be a non-zero point in the open unit ball. Let us write z =

(x1, · · · , xd), and zn := (bnx1c, · · · bnxdc) .The odometer function un(z) at rank n measures the number of walkers passing

through zn in the process of building the cluster A(⌊ωdn

d⌋). It is dened as follows :

un(z) :=

bωdndc∑i=0

σi∑t=0

1Si(t)=zn

3.2. Denitions, main results 43

Our results are the following.

Theorem 3.1. For all non-zero points of the open unit ball z, when n goes to

innity,

un(z)

n2→||z||2 − 1− 2 ln(||z||) almost surely if d = 2,

un(z)

n2→||z||2 +

2

(d− 2)||z||d−2− d

d− 2almost surely if d > 3. (3.2.1)

This theorem states that the normalised odometer converges almost surely onthe open unit disc except at the origin. Figure 3.1 presents a simulation of thedierence between un

n2 and its limit when d = 2.Remark : In [7], the authors estimate the time it takes to build a cluster of

radius n, that is to say the total number of steps done by random walks during theconstruction of the cluster. The authors count these steps by estimating the numberof steps for each given random walk. Our convergence result Theorem 3.1 allows usto take a dierent perspective on the problem, and count the total number of stepsas the sum over all points in the cluster.

The presentation of this dierent proof is the purpose of section 3.8.

These functions vanish when ||z|| tends to 1, and tend to innity when ||z|| tendsto 0. To go into further detail about the behavior around the origin, we introducean appropriate scaling. Consider a sequence of non-zero points yn such that yn

n

converges to 0 as n tends to innity. If we now consider the following sequence ofvalues taken by the odometer un at point yn

n :

un

(ynn

)=

bωdndc∑i=0

σi∑t=0

1Si(t)=yn

We get the following result :

Theorem 3.2. Let yn be a sequence of non-zero points such that ynn converges to 0.

If yn = a ∈ Zd,

un(ynn

) ∼ 2n2 ln(n) almost surely, if d = 2,

un(ynn

) ∼ ωdndG(0, a) almost surely, if d > 3.

If ||yn|| tends to innity, when n goes to innity,

un(ynn

) ∼ 4n2 ln

(n

||yn||

)almost surely, if d = 2,

un(ynn

) ∼ 2n2

d− 2

(n

||yn||

)d−2

almost surely, if d > 3.


Figure 3.1 On the top, the function f(z) = −||z||2 + 1 + 2 ln(||z||), and on the

bottom, the sum f(z) + un(z)n2 .

Conversely, the following theorem describes the behaviour of the odometer func-tion at points which are close to the border of the cluster. Note that whereas theproof of theorems 3.1 and 3.2 only require the rough estimates on the cluster obtai-ned by Lawler in [5], our next results rely on the much stronger bounds obtainedrecently by Asselah and Gaudillière in [1], and Jerison, Levine and Sheeld in [4].

Theorem 3.3. Let yn be a sequence of points such that ynn → 1, and suppose that

ηn = n − ||yn|| is such that there exists 0 < α < β < 1 and c ∈ R∗+ such that


1cn

α < ηn < cnβ. Then, we have the following limits in L2 :

un(ynn )

η2n

→ 1 if d = 2,

un(ynn )

η2n

→ 2d if d > 3.

Moreover, if α > 12 , the convergence is almost sure.

Note that the result for d > 3 does not extend to d = 2, even if it does not blowup. However, we will see in the proof that the computations that yield the resultsare very dierent, so that the fact that their result is dierent is to be expected.

In the next series of theorems, when d > 3, and in the case of points relativelyclose to the origin, we identify uctuations for the odometer function. When d = 2 orin the case of points within macroscopic distance of the origin however, our methoddoes not permit the identication of the uctuations (see the discussion at the endof section 3.7).

We start by studying the uctuations of the odometer at points within nitedistance of the origin :

Theorem 3.4. Choose a ∈ Zd, d > 3. When n goes to innity, the uctuations of

the odometer function at a are Gaussian. More precisely, given a ∈ Zd, let us denethe following function and random valuation :

ψd(a) = G(0, a)G(0, 0)

(3− P(τa <∞)− 1

G(0, 0)

),

F0,n(a) = n−d/2(un

(an

)− E

(un

(an

))), if d>2.

Then F0,n converges in law to a Gaussian random eld over Zd, with asymptotic

variance ωdψd(a), and asymptotic covariance :

limn→∞

Cov(F0,n(a), F0,n(b)) =

ωd (1− P(τa = τb =∞))1− P(τ ′0 = τb−a =∞)

P(τ ′0 = τb−a =∞)G(a, b)ωd −G(0, a)G(0, b),

where τz is the hitting time of z ∈ Zd for a walk starting at the origin, and τ ′0 is the

time of its rst return to the origin.

Remark 1 : The aim of theorem 3.4 is the study of the random componentof the odometer. It should be noted that this is not necessarily the biggest termafter the equivalent given in theorem 3.2. It is, however, the rst one to display arandom behaviour, and is as such of particular interest.

Remark 2 : The term G(a, b) in the covariance of F is reminiscent of the cova-riance function of the discrete Gaussian Free Field. In this case, it is heavily correctedby both additive and multiplicative terms, but still of interest as a Gaussian elddirectly linked to the iDLA model.


Further away from the origin, the uctuations still exhibit a Gaussian behaviour,with dierent variance and covariance. This is the object of our next theorem.

Theorem 3.5. Choose (yn)n∈N a sequence in Zd and α > 0 such that yn →∞ and

yn = o(n1−α). When n goes to innity, the uctuations of the odometer function at

yn are Gaussian. More precisely, given a ∈ Zd, let us dene if d > 2 the following

random valuation :

Fyn,n(a) =1

n

(||yn||n

)d/2−1(un

(yn + a

n

)− E

(un

(yn + a

n

)))Then Fyn,n converges in law to a Gaussian random eld over Zd, with variance

2d−2 (3G(0, 0)− 1) and the following asymptotic covariance :

limn→∞

Cov(Fyn(a), Fyn(b)) =

4

d− 2

1

1 + P(τb−a <∞)

1− P(τ ′0 = τb−a =∞)

P(τ ′0 = τb−a =∞)G(a, b)

where τz is the hitting time of z ∈ Zd for a walk starting at the origin, and τ ′0 is the

time of its rst return to the origin.

At a greater distance from the origin, our method fails because the approximationof the cluster between two discs is no longer sharp enough. We give a summation ofthe order of the uctuations and errors at the end of section 3.7.

Let us comment further on theorem 3.1, and give a heuristic of it when d > 3

based on the work of Levine and Peres ([9]) (the two-dimensional case is similarif a little more technical). In the case of the divisible sandpile model, each site ofZd contains a continuous amount of mass. A site with an amount greater than 1

can topple, that is to say keep mass 1 and distribute the rest equally between itsneighbors. With any sequence of topplings that topples each full site innitely often,the mass approaches a limiting distribution, which does not depend on the sequence.

The odometer function u for this model is dened as the total mass emittedfrom a given point. Since each neighbor y of a given point x divides mass equallybetween its 2d neighbors, the total mass received by x is 1

2d

∑y∼x u(y). If we dene

the discrete Laplacian ∆ as ∆f(x) = 12d

∑y∼x f(y)− f(x), we get that

∆u(x) = ν(x)− σ(x),

where σ and ν are the initial and nal amounts of mass at x, respectively.In our case, the initial mass ωdnd is concentrated at the origin, and the nal

mass is 1 in each fully occupied site of the cluster .Thus we can classify the sites in Zd in four categories : The origin, at which the initial mass is ωdnd, The sites that are fully occupied, at which the nal mass is 1,


A strip of partially lled sites for which un vanishes but ∆un is between 0 and1. These sites are within distance one of a fully occupied site.

All other sites, on which both un and ∆un vanish.

This leads to the following equation :

∆un(x) =

1− ωdnd if x = 0,

1 if x is in the cluster of fully occupied sites,

0 at distance greater than 2 from the cluster.

To solve this equation, we rst introduce a function γn that has discrete Lapla-cian σn − 1, in our case :

γn(z) = −||z||2 − ωdndG(0, z).

Then the odometer function un is given by lemma 3.2 of [9] as the dierencebetween the least superharmonic majorant of γn and γn :

un = sn − γn,

where sn = inff(x) such that f is superharmonic and f > γn.Recalling the notations z = (x1, · · · , xd) and zn = (bnx1c, · · · bnxdc), we get :

1

n2γn(zn)→ γ(z) := −||z||2 − 2

d− 2||z||2−d

The function γ is radial, and its particular shape makes it easy to determine itssuperharmonic majorant s, as one can see on Figure 1 :

s(z) =

− 2d−2 if ||z|| 6 1,

γ(z) if ||z|| > 1.

Since γn converges to γ, sn should converge to s. This would mean that unconverges to s− γ. This method is in fact made rigorous in [9], with the noticeabledierence that in the case considered the starting mass has a bounded density, asopposed to our case where all of the mass is started at the origin. The authors thenprove that the odometer function for the divisible sandpile model and the expectedvalue of the odometer function for the internal DLA model can be bounded in termsof one another so that they are asymptotically close, which gives a heuristic proofof the following result :

E(un(z)

n2

)→||z||2 − 1− 2 ln(||z||) if d = 2,

E(un(z)

n2

)→||z||2 +

2

(d− 2)||z||d−2− d

d− 2if d > 3.


0 1

− dd−2

s(z)

γ(z)

||z||

Figure 3.2 The functions s and γ as radial functions of z.

This analytic method could probably be rendered rigorous, but we prefer a moreprobabilistic approach based on the convergence of the cluster, which gives us analmost sure convergence result, and can yield uctuations for the odometer functionin specic cases.


The main idea of this proof is to use the fact that the cluster is very similar to adisc when n is large enough to get two inequalities framing un(z) between randomvaluations which do not depend on the shape of the cluster.

Recall that :

ϕ2(z)def= ||z||2 − 1− 2 ln(||z||)

ϕd(z)def= ||z||2 +

2

(d− 2)||z||d−2− d

d− 2if d > 3 (3.3.1)

Let us denote by En the event that the following set inequality holds :

Bn−(lnn)2 ⊂ A(⌊ωdn

d⌋) ⊂ Bn+(lnn)2 ,

Then the sub-logarithmic uctuation theorem for iDLA of Asselah and Gau-dillière (see [1], [2]) states that with probability one, En occurs for all but nitelymany n.


Let us dene the stopping time ξjk as the time at which Sj leaves a ball Bk. As aconsequence of the set inequality, we get the following inequality on stopping times :with probability one, the following statement holds for all but nitely many k :

For all j such that⌊ωd(k)d

⌋6 j 6

⌊ωd(k + 1)d

⌋,

ξj(k−1)−(ln k)2 6 σj 6 ξj

k+(ln k)2 . (3.3.2)

We get the following framing of un(z) :

n−1∑k=0


ξk−(ln k)2∑t=1

1Si(t)=zn 6 un(z) 6n−1∑k=0


ξjk+1+ln(k+1)2∑

t=1

1Si(t)=zn

This inequality holds for n large enough because ||zn|| − n||z|| is smaller thana constant. This ensures that the inequality (3.3.2) will start to hold as soon as kbecomes greater than n||z||

2 for instance. All the other terms of the sum are asymp-totically zero.

We will now show that the left and right bounds on un(z), which we will callun(z)− and un(z)+, once normalised, converge almost surely to the same function.In order to do this, we take a look at the following family of random variables :

χjk,n(z) =

ξjk∑t=1

1Sj(t)=zn

A direct application of the Markov property shows that for i > 1, there existsp0 and p depending on k, n and z such that :

P(χjk,n(z) = i) = p0pi−1(1− p)

Since we know that P(χjk,n(z) = 0) = P(τzn > ξk) where τzn is the reaching time

of zn, and that E(χjk,n(z)) = Gk(0, zn), where Gn is Green's function stopped at theball of radius n, a simple computation shows that :

p0 = P(τzn 6 ξk)

p = 1− P(τzn 6 ξk)Gk(0, zn)

This determines the parameters of the distribution χjk,n(z) follows.

We will need the following lemma which gives estimates for the stopped Greenfunctions Gn(0, z) :


Lemma 3.6. If z ∈ Bn, z 6= 0, we have

Gn(0, z) =2

ω2ln

n

||z||+O(

1

||z||) +O(

1

n) if d = 2, and

Gn(0, z) =2

d− 2

1

ωd(||z||2−d − n2−d) +O(||z||1−d),

where the O are uniform on the unspecied variables. We also have the following

estimates at z = 0 :

Gn(0, 0) =2

ω2ln(n) +O(1) if d = 2, and

Gn(0, 0) = G(0, 0) + o(1).

This lemma is due to Lawler and can be found in section 1.6 of [6].

We will now prove that 1n2un(z) converges almost surely to the limit of its mean

value. We rst compute the mean value of un(z)−, then we will bound its variance.

Consider the left side of (3.3) :

un(z)− =n−1∑k=0

bωd(k+1)dc∑j=bωdkdc

ξjk−(ln k)2∑t=1

1Sj(t)=zn

un(z)− =n−1∑k=0


χjk−(ln k)2,n

(z)

E(un(z)−) =n−1∑k=0


Gk−(ln k)2(0, zn)

=n−1∑

k=b||z||nc(⌊ωd(k + 1)d

⌋−⌊ωdk

d⌋)Gk−(ln k)2(0, zn)

(3.3.3)

Let us consider separately the case d = 2. In that case, we have :

E(un(z)−) =n−1∑

k=b||z||nc(⌊ω2(k + 1)2

⌋−⌊ω2k

2⌋)

2

ω2

(lnk − (ln k)2

n||z||+O(

1

n)

)

E(un(z)−) =

n−1∑k=b||z||nc

(4k ln

k

n||z||

)+O

(n(lnn)2

)1

n2E(un(z)−) =

1

n

n−1∑k=b||z||nc

(4k

nln

k

n||z||

)+O

((lnn)2

n

)


We recognize a Riemann sum, thus :

1

n2E(un(z)−) →

∫ 1

||z||4t ln(

t

||z||)dt = ϕ2(z)

When d > 3, we get instead the following estimate for E(un(z)−) :

n−1∑k=b||z||nc

(⌊ωd(k + 1)d

⌋−⌊ωdk

d⌋)

2

d− 2

1

ωd((n||z||)2−d − (k − (ln k)2)2−d) +O(n)

E(un(z)−) =n−1∑

k=b||z||nc

(2d

d− 2kd−1((n||z||)2−d − (k)2−d)

)+O(n)

1

n2E(un(z)−) =

1

n

n−1∑k=b||z||nc

(2d

d− 2((k

n)d−1||z||2−d − k

n)

)+O(n−1)

1

n2E(un(z)−) →

∫ 1

||z||

2d

d− 2(td−1||z||2−d − t)dt = ϕd(z)

The same arguments show that 1n2E(un(z)+) has the same limit as its counter-

part.We will now study the variance of these variables. Let us rst state the following

lemma :

Lemma 3.7. The random valuations χjk,n(z) have the following variance, for all

j, k, n ∈ N, z ∈ Zd provided that ||z||n 6 k :

Var(χjk,n(z)) = Gk(0, zn)Gk(zn, zn)

(3− 1

Gk(zn, zn)− P(τzn 6 ξk)

)The proof of this lemma is a straightforward calculation which relies only on the

knowledge of the law of χjk,n(z).We will need bounds for Gk(zn, zn). Provided that ||z||n > k, zn is in the ball of

radius k centered at the origin, and it has at least one neighbor in this same ball,which proves that

Gk(zn, zn) > 1 +1

4d2.

When d > 3, we can use the simple upper bound given by

Gk(zn, zn) 6 G(0, 0) <∞.

However, when d = 2, we will use the following bound : Gk(zn, zn) 6 Gn+k(0, 0).This bound comes from the fact that a random walk started at zn will exit the ballof radius k centered in 0 before it exits the ball of radius n+k centered in zn. Whenk 6 n, we will even use Gk(zn, zn) 6 G2n(0, 0).


We are now ready to bound the variance of un(z)−.

Var

(un(z)−

n2

)=

1

n4

n−1∑k=0

bω2(k+1)2c∑j=bωdk2c

Var(χjk−(ln k)2,n

(z))

61

n4

n−1∑k=0

bω2(k+1)2c∑j=bωdk2c

Gk−(ln k)2(0, zn)Gk−(ln k)2(zn, zn)

(3 +

1

Gk−(ln k)2(zn, zn)

)

When d > 3, we compute the exact value of the variance, since it will be usefulin the following proofs.

n4Var

(un(z)−

n2

)=

n−1∑k=0


Var(χjk−ηI(k),n(z))

=

n∑k=b||z||nc


Gk−ηI(k)(0, zn)Gk−fηI(k)(zn, zn)

(3− 1

Gk−ηI(k)(zn, zn)

)

As before, ηI(k) can be neglected everywhere it appears :

n4Var

(un(z)−

n2

)=

n∑k=b||z||nc


Gk(0, zn)Gk(zn, zn)

(3− 1

Gk(zn, zn)

)+O(n)

=

b(||z||+ε)nc−1∑k=bn||z||c


Gk(0, zn)Gk(zn, zn)

(3− 1

Gk(zn, zn)

)

+n∑

k=b(||z||+ε)nc


Gk(0, zn)Gk(zn, zn)

(3− 1

Gk(zn, zn)

)+O(n)

The rst term can be dealt with easily. There is a constant Cd depending only


on the dimension such that :

b(||z||+ε)nc−1∑k=bn||z||c


Gk(0, zn)Gk(zn, zn)

(3− 1

Gk(zn, zn)

)

61

n

b(||z||+ε)nc∑k=b(||z||)nc

Cd

(k

n

)d−1 (||z||2−d − k2−d

)+O(1)

6 εCd

(||z||2−d − (||z||+ ε)2−d

)+O(1)

The second term, on the other hand, is asymptotically equivalent to a Riemannsum :

n∑k=b(||z||+ε)nc


Gk(0, zn)Gk(zn, zn)

(3− 1

Gk(zn, zn)

)

= n2G(0, 0)2d

d− 2

(3− 1

G(0, 0)

)∫ 1

||z||+ε

(td−1||z||2−d − t

)dt+ o(n2)

Hence the result :

n2Var(unn2

)→ 2

d− 2

G(0, 0)

ωd

(3− 1

G(0, 0)

)∫ 1

||z||

(td−1||z||2−d − t

)dt

= (3G(0, 0)− 1)ϕd(z) (3.3.4)

When d = 2, taking a suited constant K2 gives us :

Var

(un(z)−

n2

)6

1

n4

n−1∑k=0

K2kGk(0, zn) ln(n)2,

which yields :

Var

(un(z)−

n2

)6

ln(n)2h(||z||)n2

,

where h is a function of ||z||.

In both cases, the sum∑

Var(un(z)−

n2

)is nite, which means un(z)−

n2 converges

almost surely to limn→∞E(un(z)−)

n2 . The same set of arguments can be used to prove

that un(z)+

n2 converges almost surely to limn→∞E(un(z)+)

n2 .

Since limn→∞E(un(z)−)

n2 = limn→∞E(un(z)+)

n2 and u−n (z) 6 un(z) 6 u+n (z) for all

n large enough, we have proved our result.



We will prove the rst part of the theorem in the case d > 3, as the proof ford = 2 is easy to derive from it.

The important point in this proof is the fact that the following framing holdsfor n large enough :

n−1∑k=blnnc



1Si(t)=yn 6

bωdndc∑i=bωd(lnn)dc

σi∑t=0

1Si(t)=yn

6n∑

k=1+blnnc


ξjk+(ln k)2∑t=1

1Si(t)=yn

This is true because the equation (3.3.2) holds as soon as k is large enough,which happens eventually since k > lnn.

The dierence to the relevant quantity can be bounded using this inequalitywhich holds for n large enough :

bωd(lnn)dc∑i=0

σi∑t=0

1Si(t)=yn 6

bωd(lnn)dc∑i=0

ξlnn−(ln(lnn)2)∑t=1

1Si(t)=yn

It is a consequence of the calculations in the proof of theorem 3.1 that theright side tends to zero almost surely once divided by nd−1.

Just like in the proof of theorem 3.1, we estimate the expected value of ourlower bound :

E

n−1∑k=blnnc



1Si(t)=yn

=n∑k=0

dωdkd−1G(||yn||) +O(nd−1)

= ωdndG(a) +O(nd−1)

The calculation for the upper bound yields the same result.

To nish the proof, bounding the variance of these random variables wouldsuce, but computing their asymptotic value is instrumental in the proof of theorem3.4. Hence, we shall compute them here in both cases d = 2 and d > 2, using lemma3.7. For more convenience, let us dene the following function :

p(yn, k) = P(τyn < ξk−(ln k)2).


The variance can now be written as follows :

Var

(un

(ynn

)−)=

n−1∑k=0



(ynn

))

=n−1∑k=0


p(yn, k)Gk−(ln k)2(yn, yn)2

(3− p(yn, k) +

1

Gk−(ln k)2(yn, yn)

).

When d = 2, we have the following asymptotics, as k tends to innity,

p(yn, k) → 1, and

Gk−(ln k)2(yn, yn) ∼ 2

πln(k)2.

A short computation yields the following equivalent for the variance :

Var(un

(ynn

))−∼ 6

πn2 ln(n)2 (3.4.1)

On the other hand, when d > 3, we have the following asymptotics, as k tends toinnity :

p(yn, k) → P(τa <∞), and

Gk−(ln k)2(yn, yn) → G(0, 0).

Since P(τa < ∞)G(0, 0) = G(0, a), this yields the following equivalent for the va-riance :

Var(un

(ynn

))−=

n−1∑k=0

dωdkd−1G(0, a)G(0, 0)

(3− P(τa <∞)− 1

G(0, 0)

)+ o(nd−1)

Hence we get :

Var

(un(ynn

)−nd

)= ωdψd(a) + o(1), (3.4.2)

where ψd(a) = G(0, a)G(0, 0)(

3− P(τa <∞)− 1G(0,0)

), as dened in Theorem

3.4.

Subsequently, the sum∑

Var

(un( ynn )

−

nd

)is nite. Since the same calculation

can be applied to u+n , we can use the same argument as in the proof of theorem


3.1 to say that almost surely, as n tends to innity,

un(ynn )

nd→ ωdG(0, a).

The proof of the second assumption in theorem 3.2 is similar to that of therst one and relies on the following estimate : for d = 2,

E

n−1∑k=0

bω2(k+1)2c∑j=bω2k2c


1Si(t)=yn

=n∑k=0

4k ln

(||yn||d−2

k − (ln k)2

)+O

(1

n

)

= 4n2 ln

(n

||yn||

)+O (n)

And for d > 3,

E

n−1∑k=0



1Si(t)=yn

=n∑k=0

2d

d− 2kd−1

(1

||yn||d−2− 1

(k − (ln k)2)d−2

)

+O

(nd−1

||yn||d−1

)

=2n2

d− 2

((n

||yn||

)d−2

− 1

2

)+O

(nd−1

||yn||d−1

)

The computations of the variances yield, for d = 2,

Var(un

(ynn

))6 J2n

2 (lnn)2 ln

(n

||yn||

),

and for d > 3,

Var(un

(ynn

))6 Jdn

2

(||yn||n

)2−d,

where J2 and Jd are suitable constants depending only on d. It follows that

Var

(un(ynn

))n2 ln

(n||yn||

) 6

J2 ln2(n)

n2 ln(

n||yn||

)Var

un(ynn

)n2 ln

(n||yn||

) 6

Jdn2

(||yn||n

)d−2

These quantities have nite sums, which proves our result.



To prove this theorem, we proceed in a similar way as before, computing theexpected value and variance of u−n and u+

n . When d = 2, we have the followingestimate of E(u−n ) :

E(u−n(ynn

))=

n∑k=n−ηn

2k ln

(k

n− ηn

)+O

(ηnn

)=

[x2

(ln

(x

n− ηn

))]nn−ηn

+O(η2n

)= −n

2

2+ n2 ln

1

1− ηnn

+n2

2

(1− 2

ηnn

+(ηnn

)2)

+O(η2n

)= ηn +O

(η2n

),

while the variance can be bounded like in the previous sections, for a suitableconstant B,

Var(u−n(ynn

))6 Bη2

n(lnn)2.

Hence, the normalised variable u−nη2nconverges to its expected value in L2. When

α > 12 , the convergence becomes almost sure as the variance Var

(u−nη2n

)6 B(lnn)2

n2α

becomes summable. Similar calculations give the same result for u+n , which yields

the result.

When d = 3, our estimates becomes :

E(u−n(ynn

))=

n∑k=n−ηn

2d

d− 2kd−1

((n− ηn)2−d − k2−d

)+O

(ηn(lnn)2

)=

2d

d− 2

[nd − (n− ηn)d

d(n− ηn)d−2− n2

2+

(n− ηn)2

2

]+O

(η2n

)= 2dη2

n +O(η2n

),

and the variance can be bounded in the following fashion :

Var(u−n(ynn

))6 B′η2

n. (3.5.1)

Once again, the normalised variable u−nη2nconverges to its expected value in L2.

When α > 12 , the convergence becomes almost sure as the variance Var

(u−nη2n

)6 B′

n2α

becomes summable. The same result holds for u+n , which nishes the proof of Theo-

rem 3.3.



The rst part of our proof will be to show that the variables which constitute thesum un

(an

)−satisfy the Lindeberg conditions, while the second part of the proof

will focus on proving that un and u−n are close enough so that they have the sameuctuations.

We recall that

un

(an

)−=

n−1∑k=0


χjk−(ln k)2,n

(an

), and

Var(χjk,n

(an

))= Gk(0, a)Gk(a, a)

(3− P(τa 6 ξk)−

1

Gk(a, a)

)From the previous sections and equation (3.4.2), we know that the sum of these

variances has the following value, when :

s2n =

n−1∑k=0



(an

))= Var

(un

(an

))= ωdψd(a)nd + o(nd)

Moreover, the random variables χjk−(ln k)2,n

(an

)can be written as the product of

a Bernoulli variable that measures whether or not Sj hits a before exiting the diskof radius k− (ln k)2, and a geometric variable that measures the number of times Sj

returns to a before exiting the disk. Since d > 3, we can bound this probability bythe probability that Sj returns to a at all. Using the lack of memory property of thegeometric distribution, we conclude that there exists a constant rd < 1 dependingonly on the dimension such that, for any t > 0, the following equation holds :

E(χjk−(ln k)2,n

(an

)21χjk−(ln k)2,n

( an

)>tsn

)6 rtsnE

((tsn + χj

k−(ln k)2,n

(an

))2)

Summing these expectations yields the following result :

s−2n

n−1∑k=0


E(χjk−(ln k)2,n

(an

)21χjk−(ln k)2,n

( an

)>tsn

)6 P (n)rtn

d/2,

where P is a polynomial in n.

This normalised sum converges to 0, so our variables satisfy the Lindeberg condi-tions. The Lindeberg-Feller theorem yields the following result, for u−n :


u−n(an

)− E

(u−n(an

))nd/2

law∼ N (0, ωdψd(a))

We now need to show that with high probability, un is close enough to u−n forthem to have the same uctuations. To do that, we use the framing of un betweenu−n and u+

n , which holds with probability one for all but nitely many n. Hence, forn large enough,

|un(an

)−− un

(an

)| 6 |un

(an

)+− un

(an

)−|

6n−1∑k=0



k−(ln k)2

1Si(t)=a

Let us dene

χjk(a) =


k−(ln k)2

1Sj(t)=a

We estimate the expected value of χjk(a) using the following lemma :

Lemma 3.8. Let Φjk be the event that the walk Sj, having reached the boundary of

Bk−(ln k)2, visits the point a ∈ Zd before exiting Bk+1+(ln k)2. Then,

P(

Φjk

)= O

(k1−d ln(k)2

).

The proof of this lemma relies on the application of the strong Markov pro-perty to the martingale G(0, S), where G is Green's function over Zd, as well asasymptotics of G.

Let us now bound the expected value of E (u+n − u−n ). For small values of k, we

will use the following bound :

E

n1/d∑k=0

ωdkd−1χk(a)

6 C1n,

where Cd is a constant depending only on d. For values of k greater than n1/d, weuse lemma 3.8, which provides the folowing bound :

E (χk(a)) 6 C2k1−d ln(k)2.


Hence, the summation yields :

E

n∑k=n1/d

ωdkd−1χk(a)

6 n∑k=n1/d

ωdθ(k) 6 C3n ln(n)2,

where C3 is a constant depending only on the dimension. Applying Markov's in-equality, with hn a sequence that goes to innity, yields

P(|u+n − u−n | > A2nhn ln(n)2

)6

C2

A2hn.

Recall that the uctuations of u−n are of order nd/2, d > 3. Hence, un is closeenough to u−n to share its uctuations :

un(an

)− E

(un(an

))nd/2

law∼ N (0, ωdψd(a)).

Since the Lindeberg-Feller Central Limit Theorem can be applied to linear com-binations of the random valuations χj

k−(ln k)2,n

(.n

)in a similar way, the resulting

eld F is indeed Gaussian.

The covariance of two of these variables can be computed in the following way :

Cov

(un

(an

), un

(b

n

))=

n∑k=1

dωdkd−1Cov

(χjk,n

(an

), χjk,n

(b

n

))

To evaluate this covariance, we start with the rst term, which we compute usingMarkov's property at the stopping time τa ∧ τb :

E(χjk,n

(an

)χjk,n

(b

n

))= (P (τa < τb ∧ ξk) + P (τb < τa ∧ ξk)) ρk(b− a)

+o (ρk(b− a)) ,

where ρk(b− a) = E(χjk,n(0)χjk,n( b−an )

).

We dene ρ as the pointwise limit of ρk. It is a consequence of Markov's propertythat ρ satises the following equation :

ρ(h) = P(τ ′0 < τh) (ρ(h) +G(0, h)) + P(τh < τ ′0) (ρ(−h) +G(0,−h))

Since both ρk and ρ are symmetric with respect to the origin, we get :

ρ(h) =1− P(τ ′0 = τh =∞)

P(τ ′0 = τh =∞)G(0, h).


Hence, when k tends to innity,

E(χjk,n

(an

)χjk,n

(b

n

))→ (1− P(τa = τb =∞))

1− P(τ ′0 = τb−a =∞)

P(τ ′0 = τb−a =∞)G(a, b)

Conversely,

E(χjk,n

(an

))E(χjk,n

(b

n

))→ G(0, a)G((0, b).

Summing these covariances, after the required normalisation, yields the resultstated in theorem 3.4.


Theorem 3.5 is proved in the same fashion as the previous theorem, hence we willonly outline the calculations that are involved. We start by computing the varianceof u−n

(ynn

). We have, as before,

un

(ynn

)−=

n−1∑k=0


χjk−(ln k)2,n

(an

), and

Var(χjk,n

(ynn

))= Gk(0, yn)Gk(yn, yn)

(3− P(τyn 6 ξk)−

1

Gk(yn, yn)

)Summing these variances yields :

n−1∑k=0



(ynn

))= Var

(u−n(ynn

))

∼ 2

d− 2(3G(0, 0)− 1)n2

(||yn||n

)2−d

Since the random variables χjk−(ln k)2,n

(ynn

)have a uniform geometric tail, the

arguments used in the previous section show that they verify the Lindeberg-Fellerconditions.

Hence, we have the following result, for u−n :

1

n

(ynn

)d/2−1 (u−n(ynn

)− E

(u−n(ynn

)))law∼ N (0, (3G(0, 0)− 1) 2

d−2)

Once again, we need to show that with high probability, un is close enough to


u−n for them to have the same uctuations. Dening as previously

χjk(yn) =

ξk+1+(ln k)2∑t=ξk−(ln k)2

1Sj(t)=yn ,

we study the expected value of this random variable, using the following corollaryto lemma 3.8, which states that its estimate holds for a larger class of points :

Corollary 3.9. Suppose that ynn → 0, k > 2||yn||. Let Φj

k(yn) be the event that the

walk Sj, having reached the boundary of Bk−(ln k)2, visits the point zn before exiting

Bk+1+(ln k)2. Then,

P(

Φjk(zn)

)= O

(k1−dθ(k)

).

To prove this corollary, let us rst consider a random walk X started at distancek − ln(k)2 from the origin and dene the following function :

Ψ(z) = E

ξk+1+(ln k)2∑i=0

1X(t)=z

Then Ψ is a discrete harmonic function on Bk+1+(ln k)2 . We use Harnack's in-

equality to conclude that its values inside a disc of radius ||yn|| are bounded in thefollowing fashion, for a suitable constant C :

Ψ(ak) 6 C

(3||yn||||yn||

)2

Ψ(0).

Hence the estimate of lemma 3.8 holds, and

P(

Φjk(zn)

)= O

(k1−dθ(k)

),

which concludes the proof.

As before, we treat the expected value of u+n − u−n in two steps, the rst step

dealing with the smaller values of k. Suppose that ||yn|| < k < 2||yn||. Then, usingthe estimates in lemma 3.6, we have

E (χk(yn)) 6 Gk+ln(k)2(0, yn)−Gk−ln(k)2(0, yn)

6 Ck1−d ln(k)2. (3.7.1)

where C is a constant depending on the dimension. Hence,

2||yn||∑k=0

ωdkd−1E (χk(yn)) 6 C||yn|| ln(n)2


For the larger values of k, we use corollary 3.9 in the following way :

n∑k=2||yn||

ωdkd−1E (χk(yn)) 6

n∑k=2||yn||

Cωd ln(k)2

6 C ′n ln(n)2

We conclude with Markov's inequality that with high probability, u+n − u−n is

suitably small. Hence, un is close enough to u−n to share its uctuations :

1

n

(||yn||n

)d/2−1 (un

(ynn

)− E

(un

(ynn

)))law→ N (0, (3G(0, 0)− 1) 2

d−2).

.

Let us now consider the eld

Fyn,n(a) =1

n

(||yn||n

)d/2−1(un

(yn + a

n

)− E

(un

(yn + a

n

))).

Once more, the Lindeberg-Feller Central Limit Theorem can be applied to linearcombinations of the random valuations, so that Fyn,n converges to a Gaussian eld.

The covariance of two of these variables can be computed in the following way :

Cov (Fyn(a), Fyn(b)) =n∑k=1

dωdkd−1Cov

(χjk,n

(yn + a

n

), χjk,n

(yn + b

n

))

To evaluate this covariance, we start with the rst term, which we compute usingMarkov's property at the stopping time τa ∧ τb :

E(χjk,n

(yn + a

n

)χjk,n

(yn + b

n

))= (P (τyn+a < τyn+b ∧ ξk) (3.7.2)

+P (τyn+b < τyn+a ∧ ξk)) (ρk,n(b− a) + o(1)) ,

where ρk,n(b− a) = E(χjk,n(0)χjk,n( b−an )

).

It is also a consequence of Markov's property that the pointwise limit ρ of ρksatises the following equation :

ρ(h) = P(τ ′0 < τh) (ρ(h) +G(0, h)) + P(τk < τ ′0) (ρ(−h) +G(0,−h))

Since both ρk and ρ are symmetric with respect to the origin, we get :

ρ(h) =1− P(τ ′0 = τh =∞)

P(τ ′0 = τh =∞)G(0, h).

We will need to evaluate the probability for a random walk starting at the origin


to hit yn + a or yn + b before exiting Bk, which we call Pyn :

Pyn = P (τyn+a < τyn+b ∧ ξk) + P (τyn+b < τyn+a ∧ ξk) .

We use the following decomposition :

P(τyn+a ∧ τyn+b < ξk) = P(τyn+a < ξk) + P(τyn+b < ξk)− P(τyn+a ∨ τyn+b < ξk)

And Markov's property to show that

P(τyn+a ∨ τyn+b < ξk) = P(τyn+a ∧ τyn+b < ξk)(P(τb−a <∞) + o(1))

Combining these two equations yields :

P(τyn+a ∧ τyn+b < ξk) =P(τyn+a < ξk) + P(τyn+b < ξk)

1 + (P(τb−a <∞) + o(1)(3.7.3)

Hence,

E(χjk,n

(an

)χjk,n

(b

n

))=

4

ωd(d− 2)

ρ(b− a)

1 + P(τb−a <∞)||yn||d−2 +O(||yn||d−1)

On the other hand, E(χjk,n

(an

))E(χjk,n

(bn

))is of order ||yn||2(d−2), so that this

term disappears after summation and normalisation.

Hence, summing these covariances, after the required normalisation, yields theresult stated in theorem 3.5.

The limits of the method

Our method for identifying uctuations relies on the fact that we know theuctuations of u−n and u+

n , which are counterparts to un built with independentrandom variables. Our method is limited by the fact that u−n and u+

n need to besuciently good approximations of un so that it will share their uctuations. Thisdierence, however, is always of order at least n ln(n)2. On the other hand, theuctuations of u−n vary greatly, depending on the position of the point and thedimension d of the space. Hence, when we consider a point within macroscopicdistance of the origin and try to identify the uctuations of un(z), the uctuationsof u−n are of order n, which is smaller than our error. Similarly, when d = 2, theuctuations around points within nite (respectively o(n)) distance of the origin,for u−n , are of order n (respectively n ln(n)), which is too small to conclude.

3.8 Back to a result of Lawler, Bramson and Grieath

The result in theorem 3.1 can be used to derive a dierent proof of an existingtime-scale result by Lawler, Bramson and Grieath. It should be noted that while

3.8. Back to a result of Lawler, Bramson and Grieath 65

this proof is somewhat simpler, it ultimately relies on stronger results obained byLawler in [5].

In [7], the authors are interested in the time it takes to build a cluster of ra-dius n, that is to say the total number of steps done by random walks during theconstruction of the cluster. They dene A(t) as the internal DLA cluster on the timescale of individual random walks. Namely, if

χ(t) = maxk such that σ1 + · · ·+ σk 6 t

then A(t) = A(χ(t)).

Theorem 3.10 (Lawler, Bramson, Grieath, 1992). Let A(t) be the internal DLA

cluster on the time scale of individual random walks, then for all ε > 0, almost

surely,

Bn(1−ε) ⊂ A(tn) ⊂ Bn(1+ε),

where

tn = nd+2 dωdd+ 2

.

While the original proof of this theorem studies the time spent by a given randomwalk inside the cluster, we have studied the time spent in a given point by all therandom walks. This leads naturally to a dierent approach of the problem, in whichwe will sum the odometer function over all the points in the cluster.

Let us call t′n the time taken to build the cluster A(n). Then

t′n =∑

z:nz∈Zd∩A(n)

un(z)

Then for all n, ∑z, nz ∈Bn−(lnn)2

un(z) 6 t′n 6∑

z, nz ∈Bn+(lnn)2

un(z)

We can compute the following asymptotics, as n tends to innity :

∑z, nz ∈Bn−(lnn)2

un(z) =

n∑k=1

dωdkd−1un

(k

n

)+ o(nd+2)

= d ωdn2

n∑k=1

kd−1f

(k

n

)+ o(nd+2), almost surely.

where f is dened by (3.2.1). Hence :

∑z:nz∈Bn−(lnn)2

un(z) = d ωdnd+2

∫ 1

0xd−1f(x)dx+ o(nd+2)

= nd+2

(d ωdd+ 2

)+ o(nd+2)

66 Bibliographie

The same asymptotics hold for∑

z, nz ∈Bn+(lnn)2un(z), so that we have, almost

surely,

limn→∞

t′nnd+2

=d ωdd+ 2

.

Bibliographie


arXiv :1009.2838, 2010.





[6] G.F. Lawler. Intersections of random walks. Birkhauser, 1996.

[7] G.F. Lawler, M. Bramson, and D. Grieath. Internal diusion limited aggrega-tion. The Annals of Probability, pages 21172140, 1992.

[8] L. Levine and Y. Peres. Strong spherical asymptotics for rotor-router aggregationand the divisible sandpile. Potential Analysis, 30(1) :127, 2009.


Chapitre 4

The Limiting Shape for DriftedInternal Diusion Limited

Aggregation is a True Heat Ball

We build the iDLA cluster using drifted random walks, and study the limitingshapes they exhibit, with the help of sandpile models. For constant drift, the norma-lised cluster converges to a canonical shape S, which can be termed a true heat ball,in that it gives rise to a mean value property for caloric functions. The existence andboundedness of such a shape answers a natural yet open question in PDE theory.


4.2 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Unfair divisible sandpile . . . . . . . . . . . . . . . . . . . . . 70

4.3.1 Denitions and notations . . . . . . . . . . . . . . . . . . . . 70

4.3.2 Convergence and abelian property . . . . . . . . . . . . . . . 72

4.3.3 Limiting shape . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.4 Regularity of D . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4 Drifted iDLA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.2 Limiting shape . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.3 A more natural class of drifted random walks . . . . . . . . . 98

4.5 Properties of the cluster . . . . . . . . . . . . . . . . . . . . . 99

4.5.1 Bounds on the cluster . . . . . . . . . . . . . . . . . . . . . . 100

4.5.2 Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.1 Introduction

The internal Diusion Limited Aggregation (iDLA) model was rst introducedby Diaconis and Fulton in [10] and gives a protocol for building a random set re-cursively. At each step, the rst vertex visited outside the cluster by a random walkstarted at the origin is added to the cluster.

68 Chapitre 4. The Limiting Shape of Drifted iDLA : a True Heat Ball

The question of the limiting shape of this model is a well-studied one. Lawler,Bramson and Grieath [21] were the rst to identify the limiting shape of the model,in the case of symmetric random walks, as a Euclidean ball. Their result was latersharpened by Lawler [20], and recently became of renewed interest with the simul-taneous works of Asselah and Gaudillère ([2], [3]) and Jerison, Levine and Sheeld([16], [17], [18] ).

Still using simple random walks, Peres and Levine introduced in [23] a methodfor identifying limiting shapes other than a Euclidean ball, in particular the limitingshape of multiple source models.

All these results are proved in the framework of simple random walks, and otherrandom walks did not appear in the literature until Blachère's article [5]. In thispaper, the iDLA model is studied for centered random walks, and convergence to-wards the ball of a specied norm is proved under moment conditions on the randomwalks. At the end of the paper, the case of drifted random walks is studied, and alimiting shape is found in the one-dimensional case. The author then conjecturesthe existence of a limiting shape for all dimensions. The initial idea that the limitingshape should be a level line of the Green's function, as it is the case in several typesof groups (see [7], [6]) is disproved.

In this paper, we present a limiting shape result for a simple class of driftedrandom walks. The limiting shape of the normalised cluster is characterised as atrue heat ball because it gives rise to a mean-value property for caloric functions.The existence of such a bounded shape is an open problem in PDE theory (see [14]),for which our convergence provides a proof inherited from the eld of random walks.

The main result in this paper is the following :

Theorem 4.1. Let An be the normalised drifted internal diusion limited aggrega-

tion cluster. Then there exists a set D ⊂ Rd−1 × R+ with the following properties :

1. Almost surely, An converges towards D with respect to the Hausdor distance.

2. Let φ be a C∞ function of time and space such that

1− p2(d− 1)

∆φ+ p∂φ

∂t= 0.

Then the following mean value property holds :∫Dφ(z, t)d(z, t) = |D|φ(0).

3. The set D is bounded in time and space.

The rst point of the theorem states our convergence result for the normalisedaggregate. The model admits a deterministic limiting shape whose properties aredetailed in points 2 and 3. The second point states a mean value property for caloricfunctions, using the shape D, which justies our use of the term true heat ball.The last point provides an answer to the as yet open problem of nding a such amean value property on a bounded set. In a nutshell, the normalised iDLA cluster

4.1. Introduction 69

converges almost surely to a bounded true heat ball. Figure 4.1 is a simulation ofthis shape, obtained with 500 000 particles.

Figure 4.1 Drifted iDLA aggregate with 500 000 particles and p = 0.2

Our proof follows the general idea of Levine and Peres, whose method for ndinglimiting shapes can be translated in our context. First, we introduce an equivalent ofthe divisible sandpile model (for a denition and convergence result of the originalmodel, see [22]), which we call the unfair divisible sandpile, because one direction isprivileged throughout the construction of the cluster. We study this model in details,and provide a limiting shape result. We prove convergence towards an abstract shapeD, which for now is dened through a parabolic obstacle problem.

Parabolic obstacle problems are relatively frequent in the literature, and occurin a number of varied elds. First and foremost, they are studied in the contextof heat diusion, and in particular in the case of two-phase transition equations,like the Stefan problem (see [24] and [9] for a discussion of general parabolic freeboundary problems). They also appear in nance, namely in problems related to


the pricing of American put options : see for example [19], [25], [15]. In our case, wewill use very strong results in parabolic potential theory (see for example [8], [4]) toensure that the limiting shape D is smooth enough, and to characterise it as a trueheat ball.

The next step of our proof is to relate the drifted iDLA and unfair divisiblesandpile, and prove that they almost surely share the same limiting shape D. Toconclude, we use probabilistic arguments to give bounds on the drifted iDLA cluster,which are in turn used to bound D, thus proving the convergence of our normalisedcluster towards a bounded true heat ball.

4.2 Heuristics

In this section, we will motivate the introduction of the divisible sandpile modelfor the study of the iDLA model. We consider the odometer function introducedin [23], that measures the total number of particles emitted from point x ∈ Zdthroughout the construction of the cluster, counted with repetitions.

A given point will then start with a certain number of particles (n if it is theorigin, 0 otherwise, in our situation), receive new particles during the constructionof the cluster, and end with a new number of particles (1 if it eventually lies insidethe cluster, 0 otherwise).

Of all the particles that passed through a given point x, it seems natural thata proportion travelled to each of the reachable neighbors of x, and that proportionshould in some sense be close to the transition probability from x to that particularneighbor.

Assuming that the same holds for each of the neighbors of x, we get a tentativelocal equation : ∑

y∼xp(y, x)u(y)− u(x)

?= ν(x)− σ(x),

where σ(x) and ν(x) are the initial and nal amounts of mass at point x, respectively.While this reasoning is awed because it assumes independence of correlated

quantities (among other problems), it motivates the introduction of a new model inwhich such a local relation holds. The unfair divisible sandpile model will play thisrole in our case.

4.3 Unfair divisible sandpile

4.3.1 Denitions and notations

We introduce the unfair divisible sandpile model to be the drifted counterpartof the divisible sandpile model dened in [22]. Consider a continuous distributionof mass on Zd, with nite total mass and bounded support. A lattice site is full ifit has mass at least 1. Any full site can topple by keeping mass 1 for itself, anddistributing the excess mass among its neighbors. While in the classical divisible

4.3. Unfair divisible sandpile 71

sandpile model, the mass is split equally among neighbors, in our model, it will bedistributed proportionally to the step distribution of the drifted random walk S.

At each time step, a full site is toppled. When we let the time go to innity, ifevery full site is toppled innitely often, the mass converges to a limiting distributionin which each site has mass less than 1. This is the object of lemma 4.2.

Just like the divisible sandpile model, our unfair version is abelian, in the follo-wing sense : while individual toppling do not commute, the limiting distribution ofmass does not depend on the order of the topplings, provided that they are done inan appropriate fashion, which is to say that each site that becomes full in the courseof the topplings is then toppled innitely often.

We prove this abelian property in lemma 4.3.

A crucial tool for the study of the divisible sandpile model, be it classical orunfair, is the odometer function. We dene it as the mass emitted from x throughoutthe construction of the cluster :

u(x) = total mass emitted from x.

It is important to note that this quantity does not depend on the sequence oftopplings either, which will derive from our proof of lemma 4.3. Since the massemitted from a point is always distributed in the same fashion, we can remark thatthe mass received by x from his neighbors is the following :

mass received in x = pu(x− ed) +1− p

2(d− 1)

∑y x,y−x⊥ed

u(y)

Since we have dened u as the emitted mass, the dierence between the receivedmass and u at point x will be equal to the dierence between the initial and nalamounts of mass at point x. Namely,

ν(x)− σ(x) = pu(x− ed) +1− p

2(d− 1)

∑y∼x,y−x⊥ed

u(y)− u(x),

where σ and ν are the initial and nal amounts of mass. We split the quantity u soas to make sense of this equation :

ν(x)− σ(x) = pu(x− ed) +1− p

2(d− 1)

∑y∼x,y−x⊥ed

u(y)− u(x)

= p (u(x− ed)− u(x)) +1− p

2(d− 1)

∑y∼x,y−x⊥ed

(u(y)− u(x))

= −p (u(x)− u(x− ed)) + (1− p)∆u(x), (4.3.1)

where ∆ is the discrete Laplacian taken over the rst d−1 coordinates of Zd, whichis to say that : ∆u(x) = 1

2(d−1)

∑y x,y−x⊥ed (u(y)− u(x)).

We now dene a discrete operator K that sums up this operation, and will play


an important role in our proofs :

Kf(x) = (1− p)∆f(x)− p (f(x)− f(x− e1)) .

We can now restate equation (4.3.1) in the following way :

ν(x)− σ(x) = Ku(x)

Given the nature of K, and since we will be inclined to consider its continuouscounterpart, we will transform notations when we pass to the continuous : the rstd− 1 will correspond to the space coordinates, while the last coordinate will corres-pond to the time coordinate t. The continuous counterpart of K will be the followingoperator :

Kf(x, t) =1− p

2(d− 1)∆f(x, t)− p∂f

∂t(x, t),

which is well known as the heat operator. Hence, we will henceforth call K the dis-crete heat operator, and dene a discrete caloric (respectively supercaloric) functionas a function f : Zd → R such that ∀z ∈ Zd,Kf(z) = 0 (respectively Kf(z) 6 0).

The sense in which K will be the continuous counterpart of K will be a non-trivialpoint in our proof, and will give rise to the non-standard normalisation we introducein section 4.3.3.1.

4.3.2 Convergence and abelian property

We will now state the convergence result and abelian property for the unfairdivisible sandpile model. Let us rst dene a toppling scheme. A toppling scheme

T for a conguration ν0 is an innite sequence of indexes in Zd in which each sitethat is initially full or becomes full through the realisation of previous topplingsappears innitely often in the remainder of the sequence. We will denote by νTkthe distribution of mass after the toppling of the rst k sites listed in T (in thespecied order), and uTk (x) the mass emitted from x up to the k-th toppling. Weomit superscripts when only one toppling scheme is involved.

In addition, a toppling scheme will be called legal if it only tries to topple fullsites.

Lemma 4.2. Consider an initial conguration ν0 with a nite total amount of mass

M and bounded support S, and a legal toppling scheme T for this distribution. Then

as k tends to innity, uk and νk tend to limits u and ν. The limiting conguration

ν is such that ∀x ∈ Zd, ν(x) 6 1. Moreover, these limits satisfy the following mass

relation :

ν = ν0 +K(u).

Proof : Let B be a ball centered at the origin big enough to contain all pointswithin graph distance M of S. We remark that any point that if νk(x) > 0 at onepoint in the construction of the cluster, it means either that x was initially in thesupport of ν0 or that it has received mass from a neighbor with mass greater that


one. Since the same reasoning can be applied to this particular neighbor, it meansthat all points that have positive mass must be within graph distanceM of S. Hence,νk is supported in B.

We introduce the weight function :

Wk =∑x∈Zd

νk(x)((1− p)(x2

1 + · · ·+ x2d−1) + pxd

).

Note that W can be negative, which does not present any problem as the keyproperty is only that it increases through topplings.

When toppling site x at time k, the mass at point x has been modied by anamount αk(x) = νk−1(x) − νk(x) which has been transferred to the neighbors of xaccording to the toppling rule. Hence,

Wk(x)−Wk−1(x) = αk(x)1− p

2(d− 1)

∑yx,y−x⊥ed

y21 − x2

1 + · · ·+ y2d−1 − x2

d−1

+pαk(x) (xd + 1− xd)= αk(x) (4.3.2)

Since uk is the sum up to k of all the relevant αi(x), we get the following relationon weights :

Wk = W0 +∑x∈Zd

uk(x).

For every x, uk(x) is a bounded increasing sequence, so it converges to a value u(x).At all nite times k, the relation

νk(x) = ν0(x) +Kuk(x)

holds, so that the convergence of νk is a consequence of that of uk. Moreover, itslimit ν is such that ν = ν0 +Ku.

Now a point x is either never toppled, in which case we have νk(x) 6 1 for allvalues of k, or it is toppled innitely often, but then each time a toppling occursat point x, ν(x) 6 1. In both cases, the limit ν(x) has to verify ν(x) 6 1, whichconcludes the proof.

We next prove the abelian property of the unfair divisible sandpile model.

Lemma 4.3. Consider an initial conguration ν0, and two legal toppling schemes

S and T . Then the corresponding nal congurations νS and νT are equal, as are

the nal odometer functions uS and uT .

Proof : Let xk be the point at which the k-th toppling occurs in the scheme S.We will prove by induction on k that uT (xk) > uSk (xk).

Suppose that the property holds for all i < k. Then for x 6= xk, either x is nottoppled before time k in scheme S, and uSk (x) = 0, or it is, in which case we considerthe last index i for which toppling occurs at point x. Then uT (x) > uSi (x) = uSk (x).In both cases, uT (x) > uSk (x).


Since S is legal, and T is a toppling scheme,

νT (xk) 6 1 6 νSk (xk).

Hence, equations (4.2) and (4.3.2) ensure that :

KuT (xk) 6 KuS(xk).

We write the mass relations for point xk :

uT (xk)−KuT (xk) = ν0(xk)− νT (xk)

uSk (xk)−KuSk (xk) = ν0(xk)− νSk (xk).

Taking the dierence yields :

uT (xk)− uSk (xk) >1− p

2(d− 1)

∑y xk,y−xk⊥ed

(uT (y)− uSk (y)

)+p(uT (xk − ed)− uSk (xk − ed)

)> 0.

The right hand side is indeed positive since it only involves dierences for pointsdierent from xk.

Hence, we have proved by induction that for all k, uT (xk) > uSk (xk). It followsthat u2 > u1. A symmetric argument shows that u1 > u2, which concludes theproof.

4.3.3 Limiting shape

4.3.3.1 Normalisation

Since the normalisation we will use is non-standard, we introduce a specic nor-malisation function φn. The fact that this normalisation is not the same in thedirection of the drift as in the other directions plays a very important part throu-ghout the paper. It is dened as follows, for functions dened on Zd with values inR :

∀f : Zd → R, φn(f) : Rd−1 × R→ Rφn(f)(x, t) = f

((bn

1d+1 (x)ic)i∈1,...,d−1, bn

2d+1 tc

).

With a slight abuse of notation, we extend the denition of φn to subsets A of Zd :

φn(A) =

(x, t) ∈ Rd−1 × R,(

(bn1d+1 (x)ic)i∈1,...,d−1, bn

2d+1 tc

)∈ A

.


4.3.3.2 The heat equation

In the following sections, we will be required to handle the heat equation, andin particular to dene what a supercaloric function is in the continuous setting. Wewill follow exactly the denition of Evans, [11]. First, we dene the fundamentalsolution Φ(x, t) of the heat equation K(f) = 0 as follows :

Φ(x, t) =

(βπt

) d−12

e−βx2

t t > 0,

0 t < 0,

where we dene

β =p(d− 1)

2(1− p).

Note that Φ is singular at the origin, and solves K(Φ) = 0 away from the origin.

For xed x ∈ Rd−1, t ∈ R, r > 0, we dene :

E(x,t;r) =

(y, s) ∈ Rd−1 × R, s 6 t,Φ(x− y, t− s) > 1

rd−1

.

Note that the boundary of E is a level set of Φ(x − y, t − s). Point (x, t) is atthe top, center end of E.

Let f be a measurable function ; we dene the operator Ar as follows :

Ar(f)(x, t) =β

4rd−1

∫E(x,t;r)

f(y, s)|x− y|2

(t− s)2dyds.

This operator is a sort of mean value operator ; indeed, if f is a smooth solutionof the heat equation, Ar can be used to compute the value at a given point :

u(x, t) = Ar(u)(x, t).

Note that the right hand side involves only u(y, s) for times s 6 t, which isreasonable, because the value at a given time should not depend upon future valuesof u.

We now use this operator to dene supercaloric functions, as follows. Let fbe a measurable function. We say that f is supercaloric if it veries the followingproperty :

∀(x, t) ∈ Rd−1 × R,∀r > 0, f(x, t) > Ar(f)(x, t).

For a function φ that has C2,1 regularity, this is equivalent to the intuitive result :

Kφ 6 0.

Note that equation (4.3.3.2) is only a weak substitute for the mean-value pro-perty of harmonic functions ; instead of simply integrating the function on a givenshape, a kernel is used. The problem of the existence of a generalized mean value


Figure 4.2 The obstacle function γ(x, t).

property for caloric functions, which is a consequence of Theorem 4.1, is discussedin [14].

4.3.3.3 Main result

The following theorem provides a result for the limiting shape of the unfairdivisible sandpile aggregate under the proper normalisation.

We run the unfair divisible sandpile on Zd with the following initial mass con-guration :

σn(x) = nδ0(x),

and we intuitively dene the unfair divisible sandpile aggregate as the set ofpoints in Zd that have positive nal mass :

Dn = x ∈ Zd, νn(x) > 0,

where νn is the nal mass conguration corresponding to the inital mass distributionµn.

Recall that β = p(d−1)2(1−p) , and dene the following quantities :

γ(x, t) = t− ||x||2 − 1

p

(β

πt

) d−12

exp

(−β ||x||

2

t

),

s(x, t) = infh(x, t)|h(x, t) is supercaloric and h > γ,u(x, t) = s(x, t)− γ(x, t)

D = (x, t) ∈ Rd−1 × R∗+, u(x, t) > 0 = s− γ > 0.

The function γ is plotted in Figure 4.2 when d = 2.Then, the following theorem states the convergence of the normalised unfair

divisible sandpile aggregate on compacts.

Theorem 4.4. For every compact K of Rd−1 × R∗+, the restriction to K of the


normalised unfair divisible sandpile aggregate φn(Dn) ∩K converges to D ∩K with

respect to the Hausdor distance.

We start by proving that the odometer function for the unfair divisible sandpilewith starting conguration k at the origin is given by the dierence between anobstacle function γn (for which we x Kγn) and its least supercaloric majorant.

Lemma 4.5. Start with mass n at the origin and choose γn such that Kγn(x, t) = 1

if (x, t) 6= (0, 0), and Kγn(0, 0) = 1− n. Then the odometer function un is given by

un = sn − γn, where sn is the least supercaloric majorant of γn.

Proof : We want to prove the following :

un + γn = inff(x, t)|Kf 6 0 and f > γn (4.3.3)

Let us rst remark that un + γn is indeed a supercaloric function such thatun + γn > γn.

We recall the mass equation (4.3.1) as stated in terms of the discrete heat ope-rator :

νn(x)− σn(x) = Kun(x).

Recall that the initial distribution of mass is n at the origin, while the naldistribution has values that are everywhere less than 1. Hence, un+γn is supercaloriceverywhere, and since un > 0 by denition, it is also a majorant of γn.

Let us now prove that is it the smallest of these functions. Let f be a supercaloricmajorant of γn. Then :

K (f − γn − un) = Kf −K (un + γ)

On Dn, K (un + γn) = 0, so that K (f − γn − un) 6 0. Outside Dn, un = 0, sothat we have f − γn − un = f − γn > 0.

Hence, f − γn − un is positive outside Dn and supercaloric inside Dn. Theminimum principle yields that f − γn − un > 0 everywhere, and f is indeed alwaysgreater than un + γn, which concludes the proof.

Remark : Lemma 4.5 gives us a way to nd the odometer function for theunfair divisible sandpile model, provided that we have a function that satises thefollowing conditions :

Kγn(x, t) = 1 if (x, t) 6= (0, 0), and

Kγn(0, 0) = 1− n. (4.3.4)

We introduce the random walk S on Zd, with the following law :

P (S(t+ 1)− S(t) = ±ei) =1− p

2(d− 1)for i = 1 · · · d− 1, and

P (S(t+ 1)− S(t) = ed) = p.

(4.3.5)


Our function γn is easily written in terms of the random walk S. Dene g(x, y)

as the expected number of times that S, started at x, visits y. Then consider thefunction :

γn(z) = zd −d−1∑i=1

z2i − ng(0, z).

(4.3.6)

It is easy to check that this function veries the conditions (4.3.4).In order to be able to use this function, we will need to evaluate the normalisation

of Green's function, which is the object of the following lemma :

Lemma 4.6. For z ∈ Zd, Dene gn(0, z) = ng(0, z). Then the normalised version

of gn, namely n−2d+1φn(gn), converges uniformly on compacts of Rd−1×R∗+ towards

the function

G(x, t) = −1

p

(β

πt

) d−12

exp

(−β ||x||

2

t

).

To prove this lemma, our rst step is to couple our random walk S with therandom walk S, whose steps are the sum of those of S between two jumps in thedirection of the drift. In other terms, the law of the increments of S is that of thelast position of simple random walk killed at a geometric time of parameter p.

We calculate the covariance matrix of the walk S, so that we can estimate itsposition at a given time using the local central limit theorem (see [26], P9).

Let us denote Pk(0, z) = P(Sk = z

). Then applying the local central limit

theorem, we get that for all (x, t) ∈ Rd−1 × R∗+,

nd−1d+1 (2πt)

d−12 P

bn2d+1 tc

(0, bn

1d+1xc

)→n→∞ (2β)

d−12 exp

(−β||x||

2

t

),

and the convergence is uniform on compacts of Rd−1×R∗+. Moreover, the dierence

between these quantities is of order n−1d+1 1√

t, see for instance [13].

The probability we estimated here is that of the event that the rst site visited

on the bn2d+1 tc-th layer in the drift direction by S is bn

1d+1xc. Hence,

g(

0,(nb

1d+1xc, bn

2d+1 tc

))=

∑z∈Zd−1

Pbn

2d+1 tc

(0, bn

1d+1xc+ z

)g((z, 0), 0) (4.3.7)

Equation (4.3.7) states that in order to get the actual number of visits to

bn1d+1xc, we have to consider the sum of the hitting probabilities of bn

1d+1xc + z,

where z ∈ Zd−1, multiplied the number of visits to 0 of the drifted random walk

started at point (z, 0). The hitting probabilities of bn1d+1xc+ z with z small enough

to be in range of 0 are asymptotically close to that of bn1d+1xc, so that we only have


to evaluate the following sum :∑z∈Zd−1

g((z, 0), 0) =∑

z∈Zd−1

g(0, (z, 0)),

using the fact that the origin and (z, 0) both have coordinate 0 in the drift direction.The summation yields exactly the expected time spent by the random walk S onone given layer.

4.3.3.4 Convergence of the obstacle function

Lemma 4.7. The normalised obstacle n−2d+1φn(γn) converges uniformly on com-

pacts K of Rd−1 × R towards γ.

The proof of this lemma is straightforward since the rst two terms in γn convergeuniformly to those of γ once normalised. The last term is the normalisation of Green'sfunction, the convergence of which was proved in lemma 4.6.

Hence, our normalised obstacle function n−2d+1φn (γn) converges uniformly on

compacts toward the function :

γ(x, t) = t− ||x||2 − 1

p

(β

πt

) d−12

exp

(−β ||x||

2

t

).

4.3.3.5 Parabolic Potential Theory

Before we go on, we will need a few lemmas of parabolic potential theory. Since

n−2d+1φn(gn) converges uniformly on compacts of Rd−1 × R∗+ towards the function

G(0, x) = −1

p

(β

πt

) d−12

exp

(−β ||x||

2

t

),

we want to use this convergence to nd a candidate to be the inverse of the heatoperator, which we dene in both the continuous and discrete setting. Let us dene,for z ∈ Zd and f a measurable function dened on Rd−1×R∗+ with compact support,

Gn(f)(z) = n∑

(y,r)∈Zd,r>0

g(z, y)f(n−

1d+1 y, n−

2d+1 r

),

and for x ∈ Rd−1 × R∗+,

G(f)(x) =

∫(y,r)∈Rd−1×R∗+

G (x, (y, r)) f(y, r)dydr.

The uniform convergence of the normalisation of gn towards g, together with ourestimate, will ensure the convergence of the normalisation of Gn(f) toward G(f).This is the object of the following lemma :


Lemma 4.8. Let f be a bounded function dened on Rd−1 × R∗+ with compact

support, then

|n−2d+1φn(Gn(f))−G(f)| → 0,

uniformly on compacts of Rd−1 × R∗+.

Suppose that f is bounded by M and supported on the compact K. We dene,for (y, r) ∈ K such that φn(y, r) ∈ Zd, the following error term :

αxn(y, r) =

∣∣∣∣n− 2d+1φn(g) (x, (y, r))−

∫zG(x, z)dz

∣∣∣∣ ,where the integral is taken over points z ∈ (y, r) + [0, n−

1d+1 )d−1 × [0, n−

2d+1 ).

Then we have :∣∣∣n− 2d+1φn(Gn(f))(x)−G(f)(x)

∣∣∣ 6∑(y,r)

Mαxn(y, r) (4.3.8)

where the sum is taken over points (y, r) ∈ K such that φn(y, r) ∈ Zd.Since we have a uniform bound on the dierence n−

2d+1φn(g) (x, (y, r)) −

G (x, (y, r)) that has nite integral on K, and using our knowledge of the deri-vatives of G to bound G (x, (y, r))−G (x, z) when z is close to (y, r), we can boundthis sum uniformly on K, which concludes the proof.

The idea behind the introduction of G(f) is that G is in some sense the inverseof the heat operator. In fact, we do not need such a strong result, and we are contentwith the following property :

Lemma 4.9. Let f be a positive measurable function dened on Rd−1 × R∗+ with

compact support, then G(f) is supercaloric.

We omit the proof as it is straightforward and only relies on commuting integralsigns and using the supercaloric property of G(x, y) as a function of x.

We are now ready to prove the convergence of the odometer functions in ourmodel.

4.3.3.6 Convergence of the odometer function

Let us consider sn, the least supercaloric majorant of γn. The following lemmastates a convergence result of the normalised version of sn.

Lemma 4.10. The normalised version of the least supercaloric majorant of γn,

namely n−2d+1φn(sn), converges uniformly to s, the least supercaloric majorant of γ,

on compacts of Rd−1 × R∗+.

We begin our proof by pointing out that our denition of the least supercaloricmajorant depends on the context : while sn is dened in the discrete setting, s isdened in the continuous.


Let K be a compact of Rd−1 × R∗+, and Kj an increasing sequence of compactsof Rd−1 × R∗+ such that ∪jKj = Rd−1 × R∗+, and Kj ⊂ ˚Kj+1. We also dene their

discrete counterparts Knj as the set of points (x, t) such that

(n−

1d+1x, n−

2d+1 t

)∈

Kj .

We dene the following quantities :

sKjn = inff(x, t)|Kf 6 0 globally, and f > γn on Kn

j ,sKj = infh(x, t)|h(x, t) is globally supercaloric and h > γ on Kj.

Remark that both functions are increasing in j, so it is a consequence of Dini'stheorem that sKj and φn(s

Kjn ) converge uniformly on K towards s and φn(sn),

respectively, as j tends to innity.

We now write that :

|φn(sn)− s| 6 |φn(sn)− φn(sKjn )|+ |φn(s

Kjn )− sKj |+ |sKj − s|

Set j big enough so that K ⊂ Kj , and |φn(sn)− φn(sKjn )|+ |sKj − s| 6 ε on K.

It now remains to show that φn(sKjn ) converges uniformly on K towards sKj .

We will now proceed to smooth the function sKj in order to show that it is closeto a function sKj such that KsKj is small.

We rst remark that sKj is continuous on K. Indeed, as it is an inmum ofcontinuous functions, it is also upper semi-continuous. Moreover, if we dene ω(γ, r)

as the continuity modulus of γ on Kj+1, we have, on Kj ,

Ar(sKj)> Ar (γ) > γ − ω(γ, r), (4.3.9)

so that Ar(sKj)+ω(γ, r) is continuous, supercaloric, and lies above γ on Kj . Hence,

Ar(sKj)6 sKj 6 Ar

(sKj)

+ ω(γ, r).

Since γ is uniformly continuous on Kj+1, Ar(sKj)→ sKj as r → 0. Since

sKj is supercaloric, this is an increasing limit. As an increasing limit of continuousfunctions, sKj is also lower semi-continuous.

We now dene, like in [11], Appendix C, for x ∈ K, λ > 0,

sKj (x) =

∫RdsKj (y)λ−dη

(x− yλ

),

where η is dened as follows :

η(x) =

C exp

(1

|x|2−1

), |x| < 1

0, |x| > 1,

with C such that∫Rd η=1 and λ is taken suciently small to ensure the denition

of the values of sKj involved. Then sKj is still supercaloric, and if λ is small enough,


it veries |sKj − sKj | 6 ε on K.

Let us dene the following discrete function :

(∀k, l) ∈ Zd−1 × Z, qn(x, t) = sKj(n−

1d+1x, n−

2d+1 t

)− n−

52(d+1) |x|2.

Since sKj is smooth and supercaloric, it veries the partial dierential inequation :

K(sKj ) 6 0.

Moreover, since sKj is smooth, Taylor's formula yields that, on K,∥∥∥n 2d+1K

(sKj

(n−

1d+1x, n−

2d+1 t

))− K

(sKj)∥∥∥ 6 An 3

d+1 ,

where A is chosen to be the maximum of the norms of third derivatives in spacedimensions, and second derivative in time of sKj .

Hence, the smoothness and supercaloric property of sKj ensure that, for n largeenough :

Kqn(x, t) 6 0.

When n is large enough, qn is close to sKj , which is in turn close to sKj , which

is greater than γ, itself close to n−2d+1φn(γn) on Kj . To sum up, the following

inequalities hold on Kj :

φn(qn) > sKj − ε > sKj − 2ε > γ − 2ε > n−2d+1φn(γn)− 3ε.

Hence qn + 3ε is a supercaloric majorant of n−2d+1φn(γn) on Kj , so

φn(sKjn ) 6 φn(qn) + 3ε 6 sKj + 3ε 6 sKj + 4ε.

We will now prove the converse inequality.

Dene the following functions :

hnj = −K(sKjn 1Kn

j+1

),

G(hnj )(x) =

∫(y,r)∈Rd−1×R∗+

G (x, (y, r))hnj (bn1d+1 yc, bn

2d+1 rc)dydr.

We will start by proving that on Knj+1,∣∣∣KsKjn ∣∣∣ 6 1.

To see this, a discrete reasoning is necessary. Indeed, let us look at one particularpoint z, and suppose KsKjn (z) < −1. Consider the function f that coincides with

sKjn on every point but z, and adjust the value f(z) such that Kf(z) = −1. Remark

that this implies f(z) < sKjn (z). We will now prove that f is a supercaloric majorant

of γn. First, it is supercaloric at point z because Kf(z) = −1 by denition, and on

other points, since f(z) 6 sKjn (z) implies that for y 6= z, Kf(y) 6 KsKjn (z). Note


that, as an inmum of discrete supercaloric functions, sKjn (z) is itself supercaloric, so

that for y 6= z, Kf(y) 6 KsKjn (z) 6 0. The function f is also a majorant of γn, whichis true at points y 6= z by denition. At point z, we know that Kf = −1 = Kγn,so that the minimum principle applied locally between z and its neighbors to thefunction f − γn guaranties f(z)− γn(z) > 0. Hence, f is a supercaloric majorant of

γn on Knj+1, so that s

Kjn 6 f , which is a contradiction. Hence, n Kn

j ,∣∣∣KsKjn ∣∣∣ 6 1.

On the other hand, since −Gn inverses K exactly, n−2d+1 s

Kjn = Gn(hnj ) on all

non-boundary points of Knj+1.

We want to argue that G(hnj )(x) is a good approximation of n−2d+1 s

Kjn (x). We

separate the error as follows, for x ∈ Knj ,∣∣∣G(hnj )(x)− n−

2d+1 s

Kjn (x)

∣∣∣ 6 A+B

where

A 6∑

(y,r)∈Knj+1

|hnj (y, r)|αxn(y, r)

B 6 B0n− 2d+1

∑(y,r),(z,t)

sKjn (y, r)αxn(z, t)− sKjn (z, t)αxn(y, r),

where the second sum is taken over points (y, r) ∼ (z, t) such that (y, t) ∈ Knj+1,

but (z, t) /∈ Knj+1, and B0 is a suitable constant depending only on the dimension.

The term B estimates the part of the error that stems from using sKjn 1Kn

j+1rather

than sKjn in the denition of hnj . To bound A, we use equation (4.3.3.6) :

A 6∑

(y,r)∈Knj+1

αxn(y, r)

We then nd the same sum as in the proof of lemma 4.8.

Bounding B is a little trickier. We proceed in the following way :∣∣∣sKjn (y, r)αxn(z, t)− sKjn (z, t)αxn(y, r)∣∣∣ 6

∣∣∣sKjn (y, r)∣∣∣ |αxn(z, t)− αxn(y, r)|

+ |αxn(y, r)|∣∣∣sKjn (y, r)− sKjn (z, t)

∣∣∣ .We will rely on three arguments : rst, remark that both (y, r) and (z, t) are

safely away from x, because of our condition Kj ⊂ ˚Kj+1 with (y, r) and (z, t)

neighbors, a short computation shows that :

αxn(z, t)− αxn(y, r) = o(n−

2d+1

).


Secondly, it follows from the denition of sKjn and the convergence of

n−2d+1φn(γn) that on Kn

j ,

sKjn (z) 6 Cn

2d+1 ,

where C is a constant depending on Kj .

Last, we use the fact that γ is uniformly continuous on Kj to argue that the

increments of γn between two neighboring points are at most of order εn2d+1 for n

large enough, and this property is in turn transferred to sKjn (since s

Kjn (.+ ei) + ε is

a supercaloric majorant of γn).

Hence, our terms are bounded in a satisfactory way, so that A and B are asymp-

totically close. Hence, on Kj , G(hn) is uniformly close to n−2d+1 s

Kjn , which is bigger

than n−2d+1γn, which is uniformly close to γ. To sum up, on Kj ,

G(hn) > n−2d+1 s

Kjn − ε > n−

2d+1γn − ε > γ − 2ε.

Moreover, lemma 4.9 states that G(hn) is supercaloric everywhere, so that

G(hn) + 2ε > s and we can conclude that sKjn + 3ε > sKj .

The convergence of sKjn towards sKj is thus uniform on K. Recall that the

convergences of sKj towards s and sKjn towards s

Kjn are guaranteed to be uniform

by Dini's theorem, which concludes the proof.

Lemma 4.11. The normalised odometer function n−2d+1φn(un) converges towards

u = s− γ uniformly on compacts of Rd−1 × R∗+.Moreover, the function u and the family n−

2d+1φn(un) have bounded (respectively

uniformly bounded) integrals over any compact of Rd−1 × R+.

The rst statement is a consequence of the convergence properties of sn and γn,while the second relies on the fact that t − ||x||2 is a supercaloric majorant of γ(resp. γn). Therefore, for x ∈ Zd,

sn(x)− γn(x) 6 ng(0, x),

and for (x, t) ∈ Rd−1 × R∗+,

s− γ 6 1

p

(β

πt

) d−12

exp

(−β||x||

2

t

).

A quick computation nishes the proof.

4.3.3.7 Convergence of Domains

Lemma 4.12. Let K be a compact of Rd−1 × R∗+. Then, on K, the normalised

unfair divisible sandpile cluster φn (Dn) converges to D with respect to the Hausdor

distance.


Set ε > 0, and dene Dε as the inside ε-neighbourhood of D, that is to say,

Dε = x ∈ D,Bx,ε ⊂ D.

Since D is the non-coincidence set for the caloric obstacle problem with obstacleγ, s − γ is strictly positive on Dε. Since s − γ is uniformly continuous on K ∩ Dε,let us dene

β = minz∈K∩Dε

(s(z)− γ(z)) > 0.

Then, for n large enough, the uniform convergences the normalised obstacle and

majorant guarantee that |γ − n−2d+1φn(γn)| < β

2 , and |s − n− 2d+1φn(sn)| < β

2 onK ∩ D. This yields the following :

n−2d+1φn(sn)− n−

2d+1φn(γn) > s− γ − β > 0 on K ∩Dε,

so that K ∩Dε ⊂ K ∩ φn(D(n)) for all but nitely many n.

Let Dε be the outside ε-neighbourhood of D. Let (xn, tn) ∈ D(n), and suppose

that (n−1d+1xn, n

2d+1 tn) converges to (x0, t0) ∈ Rd−1 × R∗+. It is sucient to prove

that (x0, t0) ∈ Dε.

Dene the discrete cylinder segment C = B(xn, n

1d+1 ε

2

)× [btn − n

2d+1 ε

2

4 c, btnc].

On C ∩D(n), dene the following function :

wn(x, t) = un(x, t)−(|x− xn|2 − (t− tn)

)Then K(wn) = Kun − 1 > 0, so that −wn is discrete supercaloric.

Hence, wn satises the parabolic maximum principle and realises its maximumon the parabolic boundary of C ∩D(n).

Since wn(x0, t0) = un(x0, t0) > 0, and un = 0 on ∂D(n), the maximum cannotbe taken on ∂D(n).

Hence wn takes its maximum on the parabolic boundary of C, ∂pC. This set canbe decomposed as follows :

∂pC = Λ1 ∪ Λ2,

where

Λ1 = B(xn, n

1d+1

ε

2

)× btn − n

2d+ε2

4c, and

Λ2 = ∂B(xn, n

1d+1

ε

2

)×[btn − n

2d+1

ε2

4c, btnc

]Now, on Λ1, we have :

wn(x, t) 6 un(x, t)− n2d+1

ε2

4,


and, similarly, on Λ2,

wn(x, t) 6 un(x, t)− n2d+1

ε2

4.

Let (yn, rn) be the point where wn takes its maximum. Then,

un(xn, tn) = wn(xn, tn) 6 wn(yn, rn)

6 un(yn, rn)

This gives the following :

un(yn, rn) > un(xn, tn) + n2d+1

ε2

4.

Let us extract a subsequence of (yn, rn) that converges to a point (y, r). Since

n−1d+1φn(un) converges to u uniformly on compacts, the limit u(y, r) is strictly

positive, so that (y, r) ∈ D.However, point (y, r) is within distance ε of (x0, t0), so we can conclude that

(x0, t0) ∈ Dε, and D(n) ⊂ Dε.

4.3.4 Regularity of D

4.3.4.1 Dierential Equation approach

So far, we have considered u only as the limit of the discrete odometer, but it willbe useful to see that it is also the solution of the continuous version of the equationthat denes un. This is the object of the following lemma :

Theorem 4.13. The normalised limit u of the odometers solves the following partial

dierential equation in the weak (distributional) sense :

1− p2(d− 1)

∆u− p∂u∂t

= 1u>0 − δ0, (4.3.10)

where δ0 is the Dirac measure at the origin.

Consider a discrete function η that has compact support on Zd. Since un solvesthe discrete equation Kun = 1un>0 − nδ0, we have :∑

z∈Zdη(z)Kun(z) =

∑z∈Zd

η(z)1un>0 − nη(0, 0) (4.3.11)

When changing the variables so as to report the operation on η, only the sign ofthe drift coordinate is modied, so that we dene the discrete operator K as theoperator K with reversed time :

Kf(z) =1− p

2(d− 1)

∑||y−z||=1,(y−z)⊥ed

(f(y)− f(z)) + p (f(z)− f(z − ed)) .


Equation (4.3.11) can now be written in the following format :∑z∈Zd

un(z)Kη(z) =∑z∈Zd

η(z)1un>0 − nη(0, 0). (4.3.12)

Let us now consider a test function h : Rd−1 × R → R that is C∞ and hascompact support. Dene its discrete counterpart hn : Zd → R as :

hn(z1, ..., zd) = h

((n−

1d+1 zi

)i∈1,...,d−1

, n−2d+1 zd

).

Then applying equation (4.3.12) yields :

1

n

∑z∈Zd

un(z)Khn(z) =1

n

∑z∈Zd

hn(z)1un>0 − h(0, 0).

1

n

∑z∈Zd

(n−

2d+1un(z)

)(n

2d+1 Khn(z)

)=

1

n

∑z∈Dn

hn(z)− h(0, 0). (4.3.13)

Theorem 4.4 proves that φn (Dn) converges to D on compacts K ⊂ Rd−1 × R∗+,and lemma 4.11 proves that n−

2d+1φn (un) converges to u uniformly on K. Hence,

letting n go to innity in (4.3.13), the regularity of h and its derivatives, togetherwith the bounded integral property stated in lemma 4.11, ensures that we can applythe dominated convergence theorem, which yields :∫

Rd−1×R∗+u

((1− p)2(d− 1)

∆h+ p∂h

∂t

)=

∫Dh− h(0, 0),

and equation (4.3.10) holds in the distributional sense.

This characterisation of u as a solution to a parabolic free boundary problemyields numerous analytical results on the function u and the set D = u > 0. Thefollowing proposition states the most important ones for our problem :

Proposition 4.14. Let φ be a C∞ function of time and space such that

1− p2(d− 1)

∆φ+ p∂φ

∂t= 0.

Then the following mean value property holds :∫Dφ(z, t)d(z, t) = |D|φ(0).

Moreover, the limit u of the odometer is continuous in both time and space coordi-

nates, and the set ∂D has (d-dimensional) Lebesgue measure 0.

The rst part of the proposition is a direct consequence our characterisationof u as a solution to the equation (4.3.10) in the distributional sense. The second


part of the property states regularity results on such a solution. For the proof ofthis powerful result in parabolic potential theory, we refer the reader to [8]. For anin-depth study of parabolic free boundary problems and their regularity, see thedetailed book by Friedman [12].

4.4 Drifted iDLA

In this section, we prove the almost sure convergence of the normalised driftediDLA cluster towards D, the limiting shape of the unfair divisible sandpile model.For reasons of simplicity and lisibility, we will start by using a random walk whoserestriction to the direction of the drift is an increasing process.

4.4.1 The model

Let (Sj)j∈N a sequence of independent random walks on Zd, with the followinglaw :

P(Sj(t+ 1)− Sj(t) = ±ei

)=

1− p2(d− 1)

for i = 1 · · · d− 1, and

P(Sj(t+ 1)− Sj(t) = ed

)= p.

(4.4.1)

The iDLA cluster is built recursively using the random walk Sj in the followingfashion. We start with an empty set. At step j, suppose that we have built A(j−1).We launch the random walk Sj at the origin, which we let evolve until it exitsA(j − 1). The rst site visited outside the cluster is then added to the cluster.

Formally, let us dene the cluster A(n) and stopping times (σk)k∈N recursivelyin the following way :

ν1 = 0,

A(1) = 0 = S1(σ1), and∀j > 1, νj = inft > 0 : Sj(t) 6∈ A(j − 1),

A(j) = A(j − 1) ∪ Sj(σj).

A few simple facts can be stated about the cluster. It is a random increasing setthat contains the origin. We have ]A(j) = j, and given the form of the law of therandom walks, the intersection of the cluster with Zd−1 × (−N∗) is empty.

4.4.2 Limiting shape

We rst prove internal convergence of our cluster intersected with compacts ofRd−1 × R∗+. We then use this convergence, together with the fact that it yieldscontrol over a proportion of the particles arbitrarily close to 1, to prove externalconvergence as well.

4.4. Drifted iDLA 89

Theorem 4.15 (Internal convergence towards D, on compacts). Let A(n) be the

drifted iDLA cluster, and K a compact subset of Rd−1 × R∗+.

Then, for every ε > 0,

Dε ∩K ⊂ φn(A(n)) ∩K,

for all but nitely many n, almost surely.

Recall that we build the cluster using the family of independant random walks(Si)i∈1···n. For z ∈ Zd, we ene the following stopping times :

τ iz = inft > 0, Si(t) = zτ iDn = inft > 0, Si(t), /∈ Dnνi = inft > 0, Si(t) /∈ A(i− 1),

We introduce the following counting random variables :

Ln(z) =

n∑i=1

1νi<τ iz<τ iDn

Mn(z) =

n∑i=1

1τ iz<τ iDn

Nn(z) =

n∑i=1

1τ iz 6 τ iDn∧νi

Since N zn measures the number of particles that pass through z before either

adding to the cluster or leaving Dn, if Nn(z) > 0, then point z is in the clusterA(n).

Remark that Nn(z) =Mn(z)− Ln(z).

The strategy of the proof will be to study these random variables. After showingthat their expected value is dierent enough, and that they are close enough to theirrespective expected values, we will deduce that their dierence is suitably big as ntends to ∞.

First, we use the Markov property of our random walk to argue that summingover all walks that add to the cluster before exiting Dn can be re-written as a sumover points of Dn. Let us construct the following family of random walks Sy, fory ∈ Dn as follows. If there is an index i ∈ 1 · · ·n such that y = Si(νi), then wedene Sy(t) = Si(t+ νi). Remark that the aggregation rule guaranties that there isat most one such index. If there is no such index, then we dene Sy as a randomwalk started at y, with the same increments as S1 for instance, and independent ofall the other random walks.

We dene the corresponding stopping times :


τyz = inft > 0, Sy(t) = zτyDn = inft > 0, Sy(t), /∈ Dn

(4.4.2)

Then we have the following inequality :

Ln(z) =αn∑i=1

1νi<τ iz<τ iDn

6∑y∈Dn

1τyz<τyDn= Ln(z),

which holds because every term in the rst sum can be found once in the secondsum, with possible additional terms. Remark that Ln(z) is now a sum of independentindicator random variables.

Let us compute the expectations of these variables :

E (Mn(z)) = ngn,Dn(0, z)

gn,Dn(z, z)

E(Ln(z)

)=

1

gn,Dn(z, z)

∑y∈Dn

gn,Dn(y, z),

where gn,Dn is the Green function of a random walk S stopped when it exits Dn.

Dene τ←z,n,Dn the time at which S←, a random walk with opposite drift, startingat z, will exit Dn. Then we have :

cE(τ←z,n,Dn)

gn,Dn(z, z)6 E

(Ln(z)

)6 C

E(τ←z,n,Dn)

gn,Dn(z, z)

Dene fn,Dn(z) = gn,Dn(z, z)E(Mn(z)− Ln(z)

). Then,

fn,Dn(z) = gn,Dn(z, z)

n∑i=1

P(τ iz < τ iDn

)−∑y∈Dn

P(τyz < τyDn

)= gn,Dn(z, z)

∑y∈Dn

(δ0(y)n− 1)P(τyz < τyDn).

=∑y∈Dn

(δ0(y)n− 1)gn,Dn(y, z)

Hence, fn,Dn is a solution to the discrete parabolic PDE :

Kfn,Dn(z) = 1− nδ0(z) ∀z ∈ Dn,fn,Dn(z) = 0 ∀z /∈ Dn.


Recall that un, the odometer function for the unfair divisible sandpile, satisesthe same equation, on the interior of Dn. This means that fn,Dn − un is a discretecaloric function on the interior of Dn. Moreover, for z on the interior boundary ofDn, since z has at least one neighbor who never toppled, it means that the totalmass emitted from z towards this neighbor is less than 1, which means we have :

un(z) 6 2d.

Hence, for all z ∈ Zd,fn,Dn(z) > un(z)− 2d

Let us dene the following minimal value of u :

β = infx∈Dε∩K

u(x).

Since u is continuous and K is a compact, the inmum is actually a minimumand, we have β > 0.

Since n−2d+1φn(un) converges uniformly to u on K, we can choose n large enough

so that for all points z ∈ Zd such that :((n−

1d+1 zi

)i∈1···d−1

, n−2d+1 zd

)∈ Dε ∩K,

we have, for n large enough,

un(z) > n2d+1

β

2

This means the following relation on the expected values of Ln(z) andMn(z) :

E(Mn(z)− Ln(z)

)>

1

2

βn2d+1

gn,Dn(z, z)(4.4.3)

Using the bounds in equation (4.4.2), we get :

E (Mn(z)) >

1 +βn

2d+1

2Cgn,Dn(z, z)E(τ←z,n,Dn

)E

(Ln(z)

)(4.4.4)

Since

((n−

1d+1 zi

)i∈1···d−1

, n−2d+1 zd

)∈ Dε ∩ K, we have for a suitably large

constant C ′, uniformly in z,

E(τ←z,n,Dn

)6 C ′n

2d+1 .

Hence, equation (4.4.4) can be written as

E (Mn(z)) > (1 + κ)E(Ln(z)

),


where κ is a strictly positive constant.

It follows that we can choose λ > 0 such that

(1 + λ)E(Ln(z)

)< (1− λ)E (Mn(z)) ,

and apply the following large deviation principle, which holds for sums of inde-pendent indicator variables :

P (L > (1 + λ)E (L)) < 2e−cλE(L)

P (M > (1− λ)E (M)) < 2e−cλE(M)

(See [1], Cor A. 14).

Since φn(Dn) converges to D with respect to the Hausdor distance, equation

(4.4.2) guaranties E(τ←z,n,Dn

)is at least a power of n, for all z ∈ Zd such that((

n−1d+1 zi

)i∈1···d−1

, n−2d+1 zd

)∈ Dε ∩K. Hence, there is a constant c2 uniform

in z such that :E (Mn(z)) > E

(Ln(z)

)> c2n

2d+1 .

Hence, both E(Ln(z)

)and E (Mn(z)) are at least powers of n.

As a consequence, with a probability that is exponentially close to 1, uniformly

in z such that(n−

1d+1 z1···d−1, n

− 2d+1 zd

)∈ Dε ∩K, the point z lies inside A(n).

Using a simple union bound on the polynomial number of points z such that :((n−

1d+1 zi

)i∈1···d−1

, n−2d+1 zd

)∈ Dε ∩ K, the Borel-Cantelli lemma states that

almost surely, for all but nitely many n,

Dε ⊂ φn (A(n)) .

Remark : The odometer function for the iDLA model does not appear directlyin this proof. However, the function fn,Dn is closely linked to it. Hence, there is reasonto believe that the normalised odometer for the drifted iDLA model also convergestowards the dierence between the obstacle function γ and its least supercaloricmajorant, as suggested by the simulation presented in Figure 4.3.

Theorem 4.16 (External convergence towards D). Let A(n) be the drifted iDLA

cluster.

Then, for every ε > 0,

φn(A(n)) ⊂ Dε,

for all but nitely many n, almost surely.

First, we remark that the regularity of the solution to the obstacle problem issuch that the Lebesgue measure of the boundary of D is 0, as stated in proposition4.14. Together with Theorem 4.15, it means that for every ε > 0, we can nd η > 0


Figure 4.3 On the top, the obstacle function γ(x, t), and on the bottom, the sumof the obstacle function and normalised odometer. The drifted iDLA cluster can beglimpsed as the non-coincidence set.


and K such that the number of points z such that :((n−

1d+1 zi

)i∈1···d−1

, n−2d+1 zd

)∈ Dε ∩K

is more than (1− ε)n.Using the classical approach of [21], we argue that we only need control over

the remaining εn points. We will need to show that these particles do not make thecluster grow beyond distance Cε of its internal limiting shape D, with C a constantdepending only on the dimension. We will do this using a layers argument.

First, we need to argue that the particles spread tout considerably when theytravel a macroscopic distance in either time (the direction of the drift) or space(other dimensions). In the direction of the drift, this is ensured by the local centrallimit theorem. The case of other dimensions is studied in the following lemma.

For simplicity reasons, we begin by stating the desired result using a simplerandom walk in the (d− 1) space dimensions.

Lemma 4.17 (First contact point with a height set). Let Xn be a (d−1)-dimensional

simple random walk starting at the origin. Let τk be the hitting time of level k, namely

τk = inft ∈ N, ||Xt||∞ > k.Then, given x0 > 0, there exists a constant C, such that for all x, t ∈

(R∗+)2,

with x > x0,

P(τbxn

1d+1 c

= btn2d+1 c

)6 Cn−

dd+1 .

This result can be explained as follows : the probability Pn that tn2d+1 is the rst

hitting time of level xn1d+1 is the product of the probability that the value xn

1d+1

is taken at time tn2d+1 by the probability, conditioned on this fact, that this level is

never reached before :

Pn = P(τxn

1d+1

= tn2d+1

)= P

(Xtn

2d+1

= xn1d+1

)P(∀t′ < tn

2d+1 , ||Xt′ ||∞ < xn

1d+1

∣∣∣Xtn

2d+1

= xn1d+1 .

)When d = 2, a symmetry argument shows that the right-hand term is equal to

the probability, for a walk starting at the origin and conditioned to reach xn13 at

time tn23 , to stay always strictly positive (except at the origin), and never go above

2xn1d+1 .

Considering only the rst condition, our conditional probability is less than theprobability, for a walk starting at the origin and conditioned to reach xn

13 at time

tn23 , to stay always strictly positive (except at the origin).

This is a well-known problem (corresponding to the probability, when countingballots, that a candidate with xn

13 more votes than the other is always rst on the

tally when the total number of votes is tn23 ), and this probability is exactly the


following :

P(∀t′ < tn

23 , t′ > 0, Xt′ > 0

∣∣∣Xtn

23

= xn13

)=xn

13

tn23

=x

tn−

13 .

When d > 2, we can apply the same argument. Indeed, suppose that the random

walk reaches level xn1d+1 through coordinate i. Then consider the projection Xi of X

to this coordinate. It is a lazy random walk, that remains in place with probabilityd−2d−1 at each step and does a simple random walk otherwise.

For any k, the probability that ||X||∞ stays less that k can be bounded aboveby the probability that |Xi| stays less than k.

Since Xi is a lazy random walk, the number of non-zero steps it takes in time

tn2d+1 is asymptotically greater than Atn

2d+1 , where A is a suitable constant.

Hence, applying the previous result to Xi yields :

P(∀t′ < tn

2d+1 , ||Xt′ ||∞ < xn

1d+1

∣∣∣Xtn

2d+1

= xn1d+1

)6

xn1d+1

Atn2d+1

Using the local central limit theorem, we get :

P(τxn

1d+1

= tn2d+1

)6 P

(Xtn

2d+1

= xn1d+1

) x

Atn−

1d+1

6Cx

Atexp

(−||x||

2

2t

)n−

d−1d+1−1

6 C ′n−dd+1 ,

where the constant C ′ is uniform in x > x0. This concludes the proof of lemma 4.17.Let us now consider our drifted random walk. We consider its projection on

the space dimensions, with a change of the time parameter such that is a simplerandom walk. Lemma 4.17 then states that the distribution of its crossing time for

level bxn1d+1 c is spread out :

P(τbxn

1d+1 c

= btn2d+1 c

)6 Cn−

dd+1 .

We now argue our drifted random walk shares the same property, in the sense that

the point at which its trajectory crosses the level xn1d+1 is spread out. Let us dene

τk = inft ∈ N,∃i ∈ 1 · · · d−1| (S(t))i | > k. Then there is a constant C ′ uniformin x > x0 > 0, t, such that :

P((

S

(τbxn

1d+1 c

))d

= btn2d+1 c

)6 C ′n−

dd+1 .

Indeed, the position in time of the crossing point is the number of steps taken by S

during the time it takes its projection to reach the level xn1d+1 , which spreads out

its distribution even more.


We will now bound the maximum distance that one of the εn particles can reachoutside the cluster in the space dimensions.

We build the cluster and stop the particles when they exit the discrete version ofDε ∩K, so that we have a number α(n) 6 εn of particles waiting to be released onthe boundary, and we dene A(i) as the cluster when i of these particles have beenreleased. It is a well-known property of the iDLA models that the law of A(α(n))

does not depend on the order in which they are released. We will re-index thesewalks as (Si)i∈1···α(n).

Choose η0 > 0, and consider the layers of points

Hk =z ∈ Zd, bn

1d+1 η0c 6 dh(z,D) 6 bn

−1d+1 η0c+ 1

,

where dh(z,D) denotes the distance, in Zd−1, between z and set of points thatnormalise into D in the d − 1 non-drift directions (We ignore the distance in thedirection of the drift). We will study the expected value of the number of particlesin each layer Hk. Le ut us dene :

µH,k(l) = E(|Hk ∩ A(l)|

).

When the l+ 1-th random walk Sl+1 is launched from the point at which it wasstuck, consider the event that it adds to the cluster at a point of Hk. Now if Sl+1

adds to the cluster at a point of Hk, it means that Sl+1 crosses Hk−1 for the rsttime at a point of Hk−1 ∩ A(l). The distance that the random walk has to cross is

bigger than η0n1d+1 , so that using equation (4.4.2) gives a uniform bound the hitting

probability of any given point. 1 simple union bound yields the following recursiverelation, for a suitable constant C1 :

µH,k(l + 1)− µk(l) 6 C1µH,k−1(l)n−dd+1

Summing over values l, and remarking that µk−1(0) = 0, yields the followingequation :

µH,k(α(n)) 6 C1n− dd+1

α(n)−1∑l=1

µH,k−1(l)

By a simple induction argument, using the inequality∑α(n)−1

l=1 lk−1 6 α(n)k

k , wehave the following inequality :

µH,k(α(n)) 6(C1n

− dd+1

)k α(n)k

k!

6

(α(n)

kC1n

− dd+1

)kChoosing k = Kεn

1d+1 , with K suitably big, results in the fact that µH,k(α(n))


is exponentially small. Hence, by the Borel-Cantelli lemma, the normalised clustercontains no points further away from D, in the direction of the drift, than η0 +Kε.

We proceed in the same way to bound the cluster in the direction of the drift.

Choose η0 > 0, and consider the layers of points

Vk =z ∈ Zd, bη0n

−1d+1 c 6 dv(z,D) 6 bη0n

−1d+1 c+ 1

,

where dv(z,D) denotes the distance, in Z, between z and points that normalise toD, in the direction of the drift.

We dene µV,k(l) as the expected value of the number of particles settled in Vkafter launching l particles.

Let us consider a walk Si. Since all points of Vk are at least at distance η0n2d+1

from the starting point of Si, the local central limit theorem gives a uniform boundfor the probability Pn of hitting any point of Vk before all the others : for a suitableconstant C2, we have :

Pn 6 C2nd−1d+1 .

Once more, since a particle needs to cross Vk−1 on a point of A(l) in order tocontribute to A(l + 1) at a point of Vk, we have a recursive relation that is similarto that of the previous proof, and for a suited constant C3,

νV,k(l + 1)− νV,k(l) 6 C3νV,k−1(l)n−d−1d+1

νV,k(α(n)) 6 C3n− d−1d+1

α(n)−1∑l=1

νV,k−1(l).

By induction, this yields :

νV,k(α(n)) 6(C3n

− d−1d+1

)k nkk!

6

(α(n)

kC3n

− d−1d+1

)k.

Choosing k = Kεn2d+1 , with K suitably big, yields that νV,k(α(n)) is exponen-

tially small. Hence, by the Borel-Cantelli lemma, the normalised cluster containsno points at distance greater than η0 + Kε from D. This concludes the proof ofTheorem 4.16


4.4.3 A more natural class of drifted random walks

Throughout our proofs so far, we have considered the drifted random walks Sj

with the following law :

P(Sj(t+ 1)− Sj(t) = ±ei

)=

1− p2(d− 1)

for i = 1 · · · d− 1, and

P(Sj(t+ 1)− Sj(t) = ed

)= p.

(4.4.5)

A more natural drifted random walk can be dened as a walk that, at each timestep, performs a simple random walk on Zd with probability 1−p, and a step in thedirection of the drift with probability p. Such a random walk S′ has the followinglaw :

P(S′(t+ 1)− S′(t) = ±ei

)=

1− p2d

for i = 1 · · · d− 1,

P(S′(t+ 1)− S′(t) = ed

)= p+

1− p2d

, and

P(S′(t+ 1)− S′(t) = −ed

)=

1− p2d

.

In order to extend our limiting shape result to the cluster built using this randomwalk, we will see step by step how our proofs need to be modied.

First, we need a dierent version of the unfair divisible sandpile, in which themass emitted from a point during its toppling is distributed in accordance withthe new law of our random walk : a fraction 1−p

2d is sent to each neighbor, and anadditional fraction p is sent towards the neighbor in the direction of the drift.

Once again, we normalise our cluster by n1d+1 in non-drift directions, and n

2d+1

in the direction of the drift.

Remark that the new version of the discrete operator we dene still veries theminimum principle, so that the nal conguration of mass is once again denedas the lowest supercaloric majorant of a suitable obstacle function, where the termsupercaloric is dened with respect to the new discrete operator.

One suitable obstacle function is the following :

γn(z) = − d

d− 1

d−1∑i=1

z2i + zd − ng(0, z),

where g is the discrete Green function for the random walk S′.

Coupling our new drifted random walk with a random walk on Zd−1 with asuitable law, we get an convergence of the normalised obstacle function towards thefunction :

γ′(x, t) = t− d

d− 1|x|2 − 1

p

(β

πt

) d−12

exp

(−β ||x||

2

t

),

4.5. Properties of the cluster 99

where β = dp2(1−p) , that is once more uniform on compacts of Rd−1 × R∗+. We also

have a suitable bound on the error.

The next step of the proof is to adapt the continuous operator K, which becomes :

K(f) =1− p

2d∆f − p∂f

∂t.

Then, following the proof of lemma 4.10, we can prove that the normalised versionof the least supercaloric majorant of γn converges to the least supercaloric majorantof γ′, where the supercaloric functions are dened as in section 4.3.3.2, with respectto the new heat operator :

K′f =1− p

2d∆f − p∂f

∂t.

This convergence is once again a little technical to prove, but it relies only onthe precise estimates of the convergence of the Green function, and the fact thatour discrete operator is a good approximation of the continuous one for functionswith sucient regularity. Remark that the terms corresponding to a simple randomwalk in the direction of the drift become negligible because of our normalisation.

Using the same arguments as in the proof of Theorem 4.2, we can thus prove thatthe normalised cluster for the new unfair divisible sandpile model, when intersectedwith a compact, converges towards the limiting shape D∩K, where D is dened asthe non-coincidence set of γ′ and its least supercaloric majorant.

The proof of the extension of the convergence result to the iDLA cluster followsthat of Theorem 4.15 almost verbatim, and the majoration of the exterior error canbe derived from the proof of Theorem 4.16, using similar estimates on the probabilityfor a random walk of hitting any given point in a set, after covering a macroscopicdistance in any direction.

4.5 Properties of the cluster

We know that our cluster converges to a limiting shape D dened in terms of anobstacle function, however we are interested in the properties of this limiting shape.To natural questions arise, the rst being its boundedness. Indeed, our normalisationonly truly captures the behaviour of the cluster if we can prove that the limitingshape under this normalisation is bounded. Moreover, we know that the limitingshape is a true heat ball, and the existence of a bounded true heat ball is a questionof interest in PDE theory.

The second natural question concerning the shape D is that of its universality :when we change the parameters of the model, we will show that the limiting shapeis only modied through a simple variable change.

100Chapitre 4. The Limiting Shape of Drifted iDLA : a True Heat Ball

4.5.1 Bounds on the cluster

In this section, we work once again with our drifted random walk S, that performsat each step a simple random in Zd with probability 1−p, and steps in the directionof the drift with probability p. Using probabilistic arguments yields bounds on theiDLA cluster, which can in turn be transferred to the continuous shape D. Hence,our limiting shape D will be a bounded true heat ball.

4.5.1.1 Horizontal bound

Lemma 4.18 (Non-drift direction bound on the cluster). The iDLA cluster A(n),

normalised by n1d+1 , is bounded in all "non-drift" directions.

Proof : We proceed as in the proof of Theorem 4.16, and we consider the inter-section of the entire cluster with strips in successively non-drift and drift directions.

Choose x0 > 0, and dene k0 = x0n1d+1 , and Γk = (x, t) ∈ Rd−1 × R, ||x||∞ =

k0 +k In this proof, we study the expected value of the number of particles in eachcylindrical domain Γk. Dene µk(l) = E (A(l) ∩ Γk).

When the l + 1-th random walk X l+1 is launched from the origin, consider theevent that it adds to the cluster at a point of Γk. Now if X l+1 adds to the clusterat a point of Γk, it means that X l+1 crosses Γk−1 for the rst time at a point ofΓk ∩ A(l). Lemma 4.4.2 gives a uniform bound on this hitting probability, so thatwe have the following recursive relation, for a suitable constant C1 :

µk(l + 1)− µk(l) 6 C1µk−1(l)n−dd+1

Summing over values l, and remarking that µk−1(0) = 0, yields the followingequation :

µk(n) 6 C1n− dd+1

n−1∑l=1

µk−1(l)

By a simple induction argument, using the inequality∑n−1

l=1 lk−1 6 nk

k , we havethe following inequality :

µk(n) 6(C1n

− dd+1

)k nkk!

6(nkC1n

− dd+1

)kChoosing k = x1n

1d+1 , with x1 suitably big, results in the fact that µk(n) is

exponentially small. Hence, by the Borel-Cantelli lemma, the normalised cluster isalmost surely asymptotically bounded by x0 + x1.

4.5. Properties of the cluster 101

4.5.1.2 Vertical bound

Lemma 4.19 (Drift-direction bound on the cluster). The iDLA cluster normalised

by n2d+1 in the direction of the drift is bounded in the direction of the drift.

Proof : The Local Central Limit theorem gives a uniform bound for the hittingprobability of a vertical strip at a particular point : for a suitable constant C2, wehave :

P(Xtn

2d+1

= xn1d+1

)6 C2n

d−1d+1

We will follow a similar strategy as in the previous proof.

Choose t0 > 0, and dene k0 = t0n2d+1 and the vertical strips Vk = (x, t)(x, t) ∈

Rd−1 × R, t = k0 + k.We will study the number of particles in each vertical strip Vk. Let νk(l) =

E (A(l) ∩ Ck0+k) denote its expected value.Since a particle needs to cross Vk−1 on a point of A(l) in order to contribute to

A(l + 1) at a point of Vk, we have a recursive relation that is similar to that of theprevious proof, and for a suited constant C3,

νk(l + 1)− νk(l) 6 C3νk−1(l)n−d−1d+1

νk(n) 6 C3n− d−1d+1

n−1∑l=1

νk−1(l)

By induction, this yields :

νk(n) 6(C3n

− d−1d+1

)k nkk!

6(nkC3n

− d−1d+1

)kChoosing k = t1n

2d+1 , with t1 suitably big, yields the summability of νk(n).

Hence, by the Borel-Cantelli lemma, the normalised cluster is almost surely asymp-totically bounded by t0 + t1.

4.5.2 Rescaling

In this section, we will show that the various limiting shapes can be written interms of one another by rescaling dierently in the direction of the drift and in theother directions.

Lemma 4.20. Let p1 and p2 be two drift parameters in (0,∞), and let D1 (res-

pectively D2) be the limiting shape of the unfair divisible sandpile model run with

parameter p1 (respectively p2).

Then D2 is the image of D1 by a change of variables x→ µx, t→ λt.

First, we introduce the additional parameter k ∈ R∗+, which measures the quan-tity of mass sent from the origin. We will call unfair divisible model with mass k

102Chapitre 4. The Limiting Shape of Drifted iDLA : a True Heat Ball

the unfair divisible sandpile model run with initial mass kn at the origin. Remarkthat the limiting shape of this model is of course obtained from that of the originalmodel by a rescaling (with dierent coecients in the drift and non-drift directions).Moreover, the limiting shape of the model can still be obtained as the solution toour parabolic obstacle problem, the only dierence being an adjusted coecient kin the last term of the obstacle function :

γk(x, t) = t− ||x||2 − k

p

(β

πt

) d−12

exp

(−β ||x||

2

t

).

We are now ready to compare D1 and D2. Since the shape D stems from theobstacle γ, we only need to show that γ2, the obstacle function for parameter p2 canbe related to γ1,k, the obstacle function for drift p1 and mass k. Consider the twofollowing functions :

1

Cγ1,k(µx, λt) =

λ

Ct− µ2

C||x||2 − k

Cp1λd−1

2

(β1

πt

) d−12

exp

(−β1

λ

µ2

||x||2

t

),

γ2(x, t) = t− ||x||2 − 1

p2

(β2

πt

) d−12

exp

(−β2||x||2

t

),

with C a suitable constant such that

K

(λ

Ct− µ2

C|||x||2

)= −1.

We see that the constant β1 can be changed to β2 = λµ2β1, provided that k is

such that the last term of 1C γ1,k is equal to that of γ2, that is to say,

k1

Cp1λd−1

2

=1

p2.

Since 1C γ1,k(µx, λt) and γ2(x, t) now only dier by a caloric function, they give rise

to the same limiting odometer, hence to the same limiting shape. We conclude usingthe rst part of the proof to recover thatD1 andD2 are indeed images of one anotherby a transformation of the required form x→ µx, t→ λt.

4.6 Conclusion

At this point, we have proved the convergence of our two models towards alimiting shape S that solves the following PDE problem : given φ a C∞ function oftime and space such that

1− p2(d− 1)

∆φ+ p∂φ

∂t= 0.

Bibliographie 103

Then the following mean value property holds :∫Sφ(z, t)d(z, t) = |S|φ(0).

We have also proved, using probabilistic estimates on random walks, that theiDLA cluster is bounded. Moreover, the regularity of the problem enables us totranspose these bounds to S, since S is suciently dened in terms of a Hausdorlimit. This concludes the proof of Theorem 4.1.

Bibliographie

[1] N. Alon and J.H. Spencer. The probabilistic method, volume 73. Wiley-Interscience, 2008.


arXiv :1009.2838, 2010.


[4] I. Athanasopoulos, L. Caarelli, and S. Salsa. Caloric functions in lipschitzdomains and the regularity of solutions to phase transition problems. The

Annals of Mathematics, 143(3) :413434, 1996.

[5] S. Blachere. Agrégation limitée par diusion interne sur zd. In Annales de

l'Institut Henri Poincare (B) Probability and Statistics, volume 38, pages 613648. Elsevier, 2002.

[6] S. Blachère. Internal diusion limited aggregation on discrete groups of poly-nomial growth. In Random walks and geometry : proceedings of a workshop at

the Erwin Schr

"odinger Institute, Vienna, June 18-July 13, 2001, page 377. De Gruyter, 2004.

[7] S. Blachère and S. Broerio. Internal diusion limited aggregation on discretegroups having exponential growth. Probability Theory and Related Fields, 137(3) :323343, 2007.

[8] L. Caarelli, A. Petrosyan, and H. Shahgholian. Regularity of a free boundaryin parabolic potential theory. Journal of the American Mathematical Society,17(4) :827870, 2004.

[9] J. Crank. Free and moving boundary problems. Oxford University Press, USA,1987.


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[11] LC Evans. Partial Dierential Equations. AMS, 1998.

[12] A. Friedman. Variational principles and free-boundary problems. Wiley, 1982.

[13] B.V. Gnedenko, A.N. Kolmogorov, K.L. Chung, and J.L. Doob. Limit distri-

butions for sums of independent random variables. Addison-Wesley Reading,MA, 1968.

[14] A. Hakobyan and H. Shahgholian. Generalized mean value property for caloricfunctions. Survey), submitted.

[15] SD Jacka. Optimal stopping and the american put. Mathematical Finance, 1(2) :114, 1991.




[19] R.A. Kuske and J.B. Keller. Optimal exercise boundary for an american putoption. Applied Mathematical Finance, 5(2) :107116, 1998.



[22] L. Levine and Y. Peres. Strong spherical asymptotics for rotor-router aggrega-tion and the divisible sandpile. Potential Analysis, 30(1) :127, 2009.


[24] A.M. Meirmanov. The stefan problem. Novosibirsk Izdatel Nauka, 1, 1986.

[25] H. Pham. Optimal stopping, free boundary, and american option in a jump-diusion model. Applied Mathematics and Optimization, 35(2) :145164, 1997.

[26] F. Spitzer. Principles of random walk. the university series in higher mathema-tics, 1964.

Chapitre 5

Containing internal diusionlimited aggregation

In the specic case where a sharp interior bound is available, we give a generalmethod for obtaining a sharp exterior bound. Our method does not rely on esti-mates of the harmonic measure but on the information already provided by theinterior bound. The result is applied to the case of iDLA on the supercritical bondpercolation cluster, where the existence of a lower bound was proved by Shellef([14]).


5.1.1 Denition of the iDLA aggregate . . . . . . . . . . . . . . . . 107

5.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.1.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Supercritical percolation . . . . . . . . . . . . . . . . . . . . . 109

5.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.1 Introduction

The Internal Diusion Limited Aggregation (iDLA) model was rst introdu-ced by Diaconis and Fulton in [7] and gives a protocol for building a randomset/aggregate recursively. At each step, the rst vertex visited outside the currentaggregate by a random walk started at the origin is added to the aggregate. Thissimulates the growth of an aggregate of particles from the inside out. In a number ofsettings, this model is known to have a deterministic limiting shape, meaning thata random aggregate with a large number of particles has a typical shape.

On Zd, Lawler, Bramson and Grieath [13] were the rst to identify this limitingshape, in the case of symmetric random walks, as an Euclidean ball. Their resultwas later sharpened by Lawler [12], and was recently drastically improved with thesimultaneous works of Asselah and Gaudillère [2, 3] and Jerison, Levine and Sheeld[10, 11, 9].

On other graphs, current knowledge is less precise. On groups with polynomialgrowth, the existence of the limiting shape is unknown, although Blachère gave

106 Chapitre 5. Containing internal diusion limited aggregation

bounds on the cluster [5]. On nitely generated groups with exponential growth,with a suitable metric, a limiting shape result was proved by Blachère and Broerio[6]. This result was then extended by Huss to a large class of random walks ongeneral non-amenable graphs [8].

Another interesting question is how robust the limiting shape is with respect tosmall perturbations of the underlying graph ; For example, on the innite cluster ofsupercritical percolation cluster of Zd, Shellef proved a sharp inner bound for theiDLA model on the innite cluster [14]. Figure 5.1 presents the iDLA aggregate builton the supercritical bond percolation cluster.

Figure 5.1 iDLA aggregate with 6 300 particles on the supercritical percolation

cluster (p=0.6). Red points are in the aggregate, green points in the cluster. the blue

points lie outside the cluster.

To prove the existence of a limiting shape, it thus remains to show a sharp outerbound on iDLA. We provide such an outer bound, relying on Shellef's inner bound.More generally, Theorem 1 states that given a sharp inner bound on iDLA for agraph (with some regularity assumptions), a sharp outer bound exists, which provesthe almost sure convergence towards a limiting shape.

Let us mention that despite the fact that the upper bound on Zd is not intrin-sically harder than the lower bound, they invoke dierent ingredients. The lowerbound usually harnesses estimates on the Green function, while the upper boundrequires the use of upper bounds on harmonic measures. The Green function is fairlywell understood in several (random) environments such as supercritical percolation,

5.1. Introduction 107

random conductances with elliptic condition, etc... This is not the case of harmonicmeasures. For instance, we do not dispose of any bound on the harmonic measure ofthe ball of size n centered at 0 in the innite cluster of supercritical percolation. Thetechniques developed in this article allow us to bypass this diculty. Our main resultrelates an upper bound on the iDLA aggregate to a lower bound. For example, if oneknows that balls of a certain metric are contained in the aggregate, and not manyparticles are left over, then one can deduce an upper bound and shape theorem.

5.1.1 Denition of the iDLA aggregate

Let G be a graph. Let S ⊂ G be a nite subset of G. In order to dene the iDLAaggregate, rst dene adding one particle started at x to an existing aggregateS. For x ∈ G, denote by A(S;x) the iDLA aggregate obtained as follows : Letξ = (ξ(0), ξ(1), . . .) be a random walk on G started at ξ(0) = x and let tS be therst time this walk is not in S. Dene

A(S;x) := S ∪ ξ(tS) .

It is standard to consider a slightly more general process, where the growth ofthe aggregate is stopped at stopping times, e.g., upon exiting a set T . Denote byA(S;x 7→ T ) the aggregate obtained by letting a particle randomly walk from x,but pausing it if it exits T : Let ξ be a random walk on G started at x. Let tS bethe rst time this walk is not in S as above, and let tT be the rst time ξ exits T .Dene

A(S;x 7→ T ) := S ∪ ξ(tS ∧ (tT − 1)) .

Note that if x is in T , the added particle is in T . To keep track of the position of apaused particle, dene

P (S;x 7→ T ) =

ξ(tT − 1) if tT < tS ,∅ otherwise,

so ∅ means that the particle is already absorbed in the aggregate.

Given vertices x1, . . . , xk and a set T , dene A(S;x1, . . . , xk 7→ T ) to be theiDLA aggregate formed from an existing aggregate S by k particles, started atx1, . . . , xk, and paused upon exiting T . That is, dene inductively : S0 = S, Sj =

A(Sj−1, xj 7→ T ) for j ∈ 1, . . . , k, and A(S;x1, . . . , xk 7→ T ) = Sk. Again, to keeptrack of paused particles, dene P (S;x1, . . . , xk 7→ T ) to be the sequence of particlespaused in this process ; Formally, if pj = P (Sj−1;xj 7→ T ) for j ∈ 1, . . . , k, thenP (S;x1, . . . , xk 7→ T ) is the sequence (pj : pj 6= ∅). When particles are not stopped,we dene the aggregate similarly and we denote it by A(S;x1, . . . , xk).

One reason to keep track of these paused particles is the so called Abelian property


of iDLA :

A(S;x1, . . . , xk) has the same distribution as

A(A(S;x1, . . . , xk 7→ T );P (S;x1, . . . , xk 7→ T )

). (5.1.1)

The equation in distribution (5.1.1) says that in order to sample A(S, x1, . . . , xk),one can sample A(S;x1, . . . , xk 7→ T ) while pausing particles and keeping trackof them via P (S;x1, . . . , xk 7→ T ), and then restart the paused particles on theobtained aggregate A(S;x1, . . . , xk 7→ T ). For more details, see [7, 13].

We are mostly interested in n particles starting at just one point. For an integern > 0, by An(x) we denote the iDLA aggregate with n particles started at x, thatis,

An(x) = A(∅;x, . . . , x︸︷︷︸n times

).

We also focus on pausing particles according to a metric ρ. Dene the ball ofradius r around x to be Bx(r) = y : ρ(x, y) < r, and denote its size by bx(r) =

|Bx(r)|. As above, set

An(x 7→ r) = A(∅;x, . . . , x 7→ Bx(r))

andPn(x 7→ r) = P (∅;x, . . . , x 7→ Bx(r)).

5.1.2 Assumptions

We now make few assumptions. The rst two assumptions are independent of theiDLA process. Assume that the metric ρ on G satises the following assumptions :

Continuity (C) : the metric is dominated by the graph distance dG on G : thereexists c > 0 such that

ρ(x, y) ≤ cdG(x, y) ∀ x, y ∈ G.

Regular volume growth (V G) : there exist c, d > 0 such that for every n > 0,

1

crd ≤ |Bx(r)| ≤ crd ∀ x ∈ Bo(n) and n1/d3 ≤ r ≤ n.

The conditions on r and n allow to consider inhomogeneous environments.We will also assume the following assumption :

Weaker lower bound (wLB) : there exists α > 0 such that for every n > 0,

P[Bx(r) ⊂ Abx(r/α)(x 7→ r)] ≥ α ∀ x ∈ Bo(n+ r) and n1/d3 ≤ r ≤ n.

In words, with noticeable probability, when releasing order bx(r) particles the ag-gregate contains Bx(r). This assumption is easy to check for many iDLA processes.

5.2. Supercritical percolation 109

In order to prove the existence of a limiting shape, a stronger lower bound for theaggregate grown around the origin is required, as we will see in the next section.

5.1.3 Main result

Theorem 1. Let G be a graph and ρ a metric on G satisfying conditions (C), (V G)

and (wLB), there exists a constant c1 > 0 so that for any ε > 0,

P[Abo(n)(o) 6⊂ Bo((1 + ε)n) i.o.] 6 P[|Abo(n)(o 7→ n)| < bo(n)(1− c1εd) i.o.].

The previous theorem is especially useful when a lower bound is known. Indeed,although we need a statement stronger than a simple lower bound, the usual proofsof lower bounds usually yield the fact that the iDLA process (almost) lls all largeenough balls when stopped on their boundary :

Lower bound (LB) : |Abo(n)(o 7→ n)|/bo(n) converges almost surely to 1.

Corollary 2. Let G be a graph and ρ a metric on G satisfying conditions (C),

(V G), (wLB). Assume also that it satises (LB), then for every ε > 0, Abo(n)(o) ⊂Bo((1 + ε)n) for n large enough almost surely.

5.2 Supercritical percolation

The main goal of this paper is the proof of the upper bound for the innitecluster of supercritical percolation. Fix d > 0 and p > pc(Zd) and consider ω tobe the innite cluster of the percolation of parameter p on Zd, conditioned on thefact that the origin o belongs to ω. In this case, we will use for ρ the Euclideandistance. From Theorem 1, we only need to check that (C), (V G), (wLB) and (LB)

are satised : Obviously, ρ ≤ dω for every environment. Therefore (C) is satised for everyω.

Barlow proved in [4] that (V G) was satised for the distance dω for almostevery environment. A classical result of [1] easily implies that for almost everyenvironment ω, there exists c1 = c1(ω) > 0 such that

ρ(x, y) ≤ dω(x, y) ≤ c1ρ(x, y) ∀n > 0 and ∀x, y ∈ Bo(n) : c1 log n ≤ ρ(x, y) ≤ n(5.2.1)

so that (V G) is also satised for ρ with a possibly dierent constant (the resultalso follows from [4]).

In [14], Shellef proved that for any ε > 0, there exists η > 0 such that thefollowing holds for almost every environment ω : there exists c2 = c2(ω) > 0

such thatP[Bo((1− ε)n) ⊂ Abo(n)(o 7→ n)

]≥ 1− c2

nd+2

and Pp[c2 ≥ λ] ≤ e−λη for all λ > 0. All together, this implies that for almost


every environment ω, there exists c3 = c3(ω) > 0 such that

P[Bx((1− ε)r) ⊂ Abx(r)(x 7→ r) ∀x ∈ Bo(n) and n1/d3 ≤ r ≤ n

]≥ 1−c2

n.

(5.2.2)The condition (wLB) follows readily. The result in Shellef deals with the eventBo((1 − ε)n) ⊂ Abo(n)(o) (particles are not stopped at distance n), but theproof does actually imply the stronger result.

Finally, note that the comparison between distances (5.2.1) imply that

bo(n)− bo((1− ε)n) ≤ c4εdbo(n)

for some constant c4 = c4(ω) > 0 depending on the environment, so that(5.2.2) implies (LB) for almost every environment.

Together with Shellef's lower bound, we thus obtain the following limiting-shapetheorem for percolation :

Theorem 3. Fix d > 0 and p > pc(Zd). For almost every environment, the iDLA

aggregate on the innite cluster of percolation of parameter p on Zd satises :

Bo((1− ε)n) ⊂ Abo(n)(o) ⊂ Bo((1 + ε)n) ∀n large enough,

where the balls B are balls for the Euclidean norm.

5.3 Proofs

From now on, we x a graph G satisfying (C), (V G) and (wLB). Constants inthe proof always depend on the constants involved in (C), (V G) and (wLB) (i.e.c, α and d) only.

The following lemma shows that a lower bound on the aggregate implies a lo-wer bound on hitting probabilities. It is a general statement not invoking any ofthe conditions (C), (V G) or (wLB). In the following, we make a slight abuse ofnotations : ξ will denote a random walk as well as its trace.

Lemma 5.1. Let Q ⊂ B ⊂ G and x ∈ B. Let ξ be a random walk started at x and

stopped on exiting B. For any t > 0,

P[ξ ∩Q 6= ∅] > P[B ⊂ At(x 7→ B)] · |Q|/t.

The above lemma is most useful when t is chosen so that P[B ⊂ At(x 7→ B)] is oforder 1.

Let ξ1, . . . , ξt be the t independent random walks started at x and stopped onexiting B that generate the aggregate At(x 7→ B). Let J ∈ 1, . . . , t be a uniformlychosen index independent of the random walks. Consider the set Γ of j ∈ 1, . . . , tso that ξj hits Q before exiting B. Since Q ⊂ B, the inclusion B ⊂ At(x 7→ B)

5.3. Proofs 111

implies |Γ| > |Q|. Since J is independent of ξ1, . . . , ξt,

P[ξJ ∩Q 6= ∅ | B ⊂ At(x 7→ B)] > P[J ∈ Γ | B ⊂ At(x 7→ B)] > |Q|/t.

The lemma follows since the distribution of ξJ is that of a random walk started atx and stopped when exiting B.

By assumption on G, we can hence get the following hitting probability estimate.

Lemma 5.2. There exist ε, η > 0 such that for large enough n and n1/(d(d+1)) <

r < n, the following holds. Let x ∈ Bo(n) and let S ⊂ Bo(n + r) be so that |S \Bo(n)| 6 εrd. Let ξ be a random walk started at x and stopped upon exiting Bo(n+r).

Then,

P[ξ ∩

[Bo(n+ r) \ (S ∪Bo(n))

]6= ∅]≥ η.

For every path γ from inside Bo(n) to outside Bo(n + r), let y(γ) be the rstvertex on γ so that ρ(y(γ), Bo(n)) > r/2. Denote by Y the set of all y(γ) for suchpaths γ. Every path from x to outside Bo(n+r) must hits Y . By Markov's property,it suces to prove the theorem for starting points y ∈ Y . Fix y ∈ Y .

Let B = By(r/3) and Q = B \ S. By (V G) and by assumption on S,

|Q| ≥ 1

c(r/3)d − εrd ≥ 1

c(r/4)d,

with ε = 4−d/c. By (wLB) with t = by(r/(3α)),

P[B ⊂ At(y 7→ r/3)] ≥ α.

Let ξ be a random walk started at y and stopped on exiting B. Lemma 5.1 and(V G) imply that

P[ξ ∩Q 6= ∅] > α (r/4)d

cby(r/(3α))> α(α/4)d/c2 := η.

Note that r/2 ≤ ρ(y,Bo(n)) 6 r 6 r/2 + c thanks to the denition of Y and (C).Therefore, ρ(y,G\Bo(n+r)) > r−r/2−c > r/3 for n large andB ⊂ Bo(n+r)\Bo(n).We deduce

P[ξ ∩

[Bo(n+ r) \ (S ∪Bo(n))

]6= ∅]≥ P[ξ ∩Q 6= ∅].

After analyzing the behavior of a single particle, we can analyze the behaviorof the whole aggregate. The following lemma says that, with high probability, aconstant fraction of the aggregate is absorbed in a wide enough annulus.

Lemma 5.3. There exist δ > 0 and p < 1 such that for all n large enough, for all

n1/(d+1) < k < n and x1, . . . , xk ∈ Bo(n), and for all S ⊂ Bo(n),

P[|A(S;x1, . . . , xk 7→ Bo(n+ k1/d)) \ S| ≤ δk ] ≤ pk.


Let r = k1/d. Fix η, ε as in Lemma 5.2. Let ξ1, . . . , ξk be the random walks startedat x1, . . . , xk and generating the aggregate. Let k′ = bεkc 6 εrd. For j ∈ 1, . . . , k′,denote

Aj = A(S;x1, . . . , xj 7→ B(o, n+ r)).

Lemma 5.2 implies that for all j ∈ 1, . . . , k′,

P[ξj+1 ∩

[Bo(n+ r) \Aj

]6= ∅ | Aj

]≥ η

Therefore, |A(S;x1, . . . , xk 7→ Bo(n + r)) \ S| dominates a (k′, η)-binomial randomvariable. Therefore, there exists δ > 0 and p < 1 that depend only on ε, η such that

P[|A(S;x1, . . . , xk 7→ Bo(n+ r)) \ S| ≤ δk

]≤ pk.

We now turn to prove Theorem 1. The proof consists of inductively constructinga sequence of aggregates Aj by pausing the particles at dierent distances nj fromthe origin. If kj is the number of paused particles, we choose the next distancenj+1 at which we pause the particles again in terms of nj and kj . We iterate thisprocedure until there are less than n1/(d+1) paused particles.

Fix n > 0. Dene Aj , nj , Pj , kj as follows :

Let n0 = n and A0 = Abo(n)(o 7→ Bo(n)). Let P0 = Pbo(n)(o 7→ Bo(n)) and letk0 = |P0|.

For j > 0, dene

nj+1 =

nj + k

1/dj if kj > n1/(d+1),

∞ otherwise.

Let Aj+1 = A(Aj ;Pj 7→ Bo(nj+1)). Let Pj+1 = P (Aj ;Pj 7→ Bo(nj+1)) and letkj+1 = |Pj+1|.

Let J be the (random) rst time at which kJ 6 n1/(d+1). By construction, Aj = AJ+1

for any j ≥ J + 1. The Abelian property (5.1.1) guarantees that AJ+1 and Abo(n)(o)

have the same law.

By construction, AJ ⊂ Bo(nJ). Since kJ 6 n1/(d+1) and ρ is continuous (C), thekJ last particles cannot grow long arms. Formally, AJ+1 ⊂ Bo(nJ + cn1/(d+1)).

Since J ≤ n, by Lemma 5.2, for some δ = δ(α, c, d) < 1,

P[∃ 1 6 j 6 J : kj > (1− δ)jk0] 6 P[∃ 1 6 j 6 J : kj > (1− δ)kj−1] 6 npn1/(d+1)

.

This implies that with probability larger than 1− npn1/(d+1),

nJ = n+ k1/d0 + · · ·+ k

1/dJ−1 ≤ n+ k

1/d0 · 1

1− (1− δ)1/d,

and if nJ +Cn1/(d+1) > (1 + ε)n, then k1/d0 > (εn−Cn1/(d+1))(1− (1− δ)1/d). So,

Bibliographie 113

using (V G), for any ε > 0,Abo(n)(o) 6⊂ Bo((1 + ε)n)

⊂k0 > c1ε

db(o, n)∪∃ 1 6 j 6 J : kj > (1− δ)jk0

.

Using the Borel-Cantelli Lemma, we deduce the result easily.

Bibliographie

[1] P. Antal and A. Pisztora. On the chemical distance for supercritical bernoullipercolation. The Annals of Probability, pages 10361048, 1996.


arXiv :1009.2838, 2010.


[4] M.T. Barlow. Random walks on supercritical percolation clusters. The Annalsof Probability, 32(4) :30243084, 2004.

[5] S. Blachère. Internal diusion limited aggregation on discrete groups of poly-nomial growth. In Random walks and geometry : proceedings of a workshop at

the Erwin Schr

"odinger Institute, Vienna, June 18-July 13, 2001, page 377. De Gruyter, 2004.

[6] S. Blachère and S. Broerio. Internal diusion limited aggregation on discretegroups having exponential growth. Probability Theory and Related Fields, 137(3) :323343, 2007.


[8] W. Huss. Internal diusion-limited aggregation on non-amenable graphs. Elec-tronic Communications in Probability, 13 :272279, 2008.





114 Bibliographie


[14] E. Shellef. Idla on the supercritical percolation cluster. Electronic Journal of

Probability, 15 :723740, 2010.

Annexe A

Algebraic properties of thedivisible sandpile model

In this annex, we prove a few algebraic results about the divisible sandpile model.We call DS the function that maps every inital conguration with nite total massand bounded support to its nal conguration. We prove that DS is 1-Lipschitzwith respect to the total variation distance on mass congurations, and that it is,in a sense to be dened, associative and linear.

A.1 Lipschitz regularity of the divisible sandpile model

Let us dene the function DS which associates to every mass conguration onZd with nite total mass and supported on B(0,M) the nal distribution given bythe divisible sandpile model dened in [? ].

Lemma A.1. DS is 1-Lipschitz for the L1-distance.

Proof :

Choose two mass distributions µ1, µ2 , and dene :

ν1 = DS(µ1)

ν2 = DS(µ2)

µ+ = max(µ1, µ2)

µ− = min(µ1, µ2),

Then ∑y∈Zd

µ+(y)− µ−(y) =∑y∈Zd|µ2(y)− µ1(y)|

Dene ν+ = DS(µ+), ν− = DS(µ−). Then we have

|ν2 − ν1| 6 ν+ − ν−.

Indeed, suppose that ν2(y) > ν1(y).

Then, since µ+ > µ2 and DS is monotone, ν+ > ν2. Similarly, ν− 6 ν1, whichgives the result.

116 Annexe A. Algebraic properties of the divisible sandpile model

Hence, using the fact that DS preserves the total mass,∑y∈Zd|ν2(y)− ν1(y)| 6

∑y∈Zd

ν+(y)− ν−(y)

6∑y∈Zd

ν+(y)−∑y∈Zd

ν−(y)

6∑y∈Zd

µ+(y)−∑y∈Zd

µ−(y)

6∑y∈Zd

µ+(y)− µ−(y)

6∑y∈Zd|µ2(y)− µ1(y)|.

Therefore, DS is 1-Lipschitz for the L1 distance.

A.2 Associativity and linear property of the divisible

sandpile model

Lemma A.2. The operations DS and adding a mass conguration on any congu-

ration 'behave well together'. That is to say :

Let µ1, µ2 be mass congurations on Zd. Then

DS(µ1 + µ2) = DS(DS(µ1) + µ2).

In other words, starting with conguration µ1 + µ2, and stabiliing it, yields ν.Starting with conguration µ1, stabilising it into ν1, then stabilising ν1 + µ2,

yields ν.Our lemma states that ν = ν.Proof :Let z ∈ Zd be a point with mass M > 1. If m < M − 1, we dene a m-partial

toppling of z as the following operation : leave mass 1 +m at point z, send mass 1

2d(M −m− 1) to each of the 2d neighbours of z.Then for any suitable m, doing a m-partial toppling of point z then a regular

toppling gives the same nal mass distribution as simply doing a regular toppling.Let X be a sequence in Zd that visits any given point innitely often. Then

dene ν1,n as the limit of the mass distribution given by doing all possible topplingsat points X1, · · ·Xn

1 starting with conguration µ1.Then ν1,n converges pointwise to ν1.Note that ν1,n + µ2 is also the result of the sequence of µ2(Xi)-partial topplings

at points X1, · · ·Xn, with starting conguration µ1 + µ2.Since the nal mass distribution does not depend on the order of topplings, and

each point is toppled innitely often, we can rearrange the topplings in such a way

A.2. Associativity and linear property of the divisible sandpile model117

that every partial toppling is immediately followed by a regular toppling at the samepoint. This gives the following result :

DS(ν1,n + µ2) = DS(µ1 + µ2). (A.2.1)

since DS is 1-Lipschitz for the L1 distance, and since ν1,n converges pointwiseto ν1, equation (A.2.1) gives :

DS(ν1 + µ2) = DS(µ1 + µ2).

Lemma A.3. The Divisible Sandpile addition is associative : given three mass dis-

tributions µ1, µ2, µ3 on Zd,

DS(µ1 +DS(µ2 + µ3)) = DS(DS(µ1 + µ2) + µ3)

Proof : this is a direct consequence of the previous lemma :

DS(µ1 +DS(µ2 + µ3)) = DS(µ1 + µ2 + µ3) = DS(DS(µ1 + µ2) + µ3)

Lemma A.4. The operation DS also behaves well with multiplication, in the follo-

wing sense : if λ > 1,

DS(λDS(µ)) = DS(λµ)

Proof : the proof is the same as previously, doing λ-partial topplings and usinga continuity argument yields the result.Remark : combining these two results, we get :

DS(µ1 + λµ2) = DS(µ1 + λDS(µ2)),

which we call a linear property for the divisible sandpile model.

Etude du modèle de l'Agrégation Limitée par Diusion Interne

Résumé : Cette thèse contient quatre travaux sur le modèle d'Agrégation Limitéepar Diusion Interne (iDLA), qui est un modèle de croissance pour la constructionrécursive d'ensembles aléatoires.

Le premier travail concerne la dimension 1 et étudie le cas où les marchesaléatoires formant l'agrégat évoluent dans un milieu aléatoire. L'agrégat normaliséconverge alors non pas vers une forme limite déterministe comme dans le cas demarches aléatoires simples mais converge en loi vers un segment contenant l'originedont les extrémités suivent la loi de l'Arcsinus.

Dans le deuxième travail, on considère le cas où l'agrégat est formé par desmarches aléatoires simples en dimension d > 2. On donne alors des résultats deconvergence et de uctuations sur la fonction odomètre introduite par Levine etPeres, qui compte en chaque point le nombre de passages des marches ayant formél'agrégat.

Dans le troisième travail, on s'intéresse au cas où l'agrégat est formé par desmarches aléatoires multidimensionnelles qui ne sont pas centrées. On montre quesous une normalisation appropriée, l'agrégat converge vers une forme limite quis'identie à une vraie boule de chaleur. Nous répondons ainsi à une question ouverteen analyse concernant l'existence d'une telle boule bornée.

Le quatrième travail concerne le cas particulier où une borne intérieure estconnue pour l'agrégat. On donne alors des conditions susantes sur le graphe ainsique sur la nature de cette borne pour qu'elle implique une borne extérieure. Cerésultat est appliqué au cas de marches évoluant sur un amas de percolation pararêtes surcritique, complétant ainsi un résultat de Shellef.

Mots-clés : Marche aléatoire, Modèle de croissance, Théorie du potentiel para-bolique, Percolation, Problème à frontière libre, Tas de sable divisible.

On the Internal Diusion Limited Aggregation model

Abstract : This thesis contains four works on the Internal Diusion LimitedAggregation model (iDLA), which is a growth model that recursively builds randomsets.

The rst work is set in dimension 1 and studies the case where the ran-dom walks that build the aggregate evolve in a random environment. Thenormalised aggregate then does not converges towards a deterministic limitingshape as it is the case for simple random walks, but converges in law towardsa segment that contains the origin and which extremal points follow the Arcsine law.

In the second work, we consider the case where the aggregate is built by simplerandom walks in dimension d > 2. We give convergence and uctuation results onthe odometer function introduced by Levine and Peres, which counts at each pointthe number of visits of walkers throughout the construction of the aggregate.

In the third work, we examine the case where the aggregate is built using mul-tidimensional drifted random walks. We show that under a suitable normalisation,the aggregate converges towards a limiting shape which is identied as a trueheat ball. We thus give an answer to an open question in analysis concerning theexistence of such a bounded shape.

The last work deals with the special case where an interior bound is known forthe aggregate. We give a set of conditions on the graph and on the nature of thisinterior bound that are sucient to imply an outer bound. This result is appliedto the case of random walks on the supercritical bond percolation cluster, thuscompleting a result by Shellef.

Keywords : Random walk, Growth model, Parabolic potential theory, Perco-lation, Internal DLA, divisible sandpile.

etude du modèle d'agrégation limitée par diffusion ...régularité que peut atteindre un tel...

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