pinning and wetting transition for (1+1)-dimensional...

The Models Free Energy Path Results Proofs

Pinning and Wetting Transition for(1+1)-Dimensional Fieldswith Laplacian Interaction

Francesco Caravenna

[email protected]

Universita degli Studi di Padova

Seminar on Stochastic Processes

Zurich, April 30th, 2008

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 1 / 34

References

I [CD1] F. Caravenna and J.-D. DeuschelPinning and wetting transition for (1+1)-dimensional fieldswith Laplacian interaction, Ann. Probab. (to appear)

I [CD2] F. Caravenna and J.-D. DeuschelScaling limits of (1+1)-dimensional pinning models withLaplacian interaction, preprint (2008).

Outline

1. The ModelsIntroductionWetting and pinning models

2. Free Energy ResultsThe free energyThe phase transitionThe disordered case

3. Path ResultsPath resultsRefined critical scaling limit

4. Sketch of the ProofsIntegrated random walkMarkov renewal theory

Outline

The Models Free Energy Path Results Proofs Introduction

Some motivations

DNA denaturation transition at high temperature

(1+1)-dimensional model: field above an impenetrable wall

Some motivations

Energy

Entropy

The Models Free Energy Path Results Proofs Wetting and pinning models

The general wetting model

The field ϕ = ϕi1≤i≤N in the free case:

(dϕ1 , . . . , dϕN

e−HN(ϕ)

N∏i=1

+ ε · δ0(dϕi ))

I dϕ+i is the Lebesgue measure on [0,∞)

I HN(ϕ) describes the structure of the chain (to be specified)

I Zw0,N is the normalization constant (partition function)

I δ0(·) is the Dirac mass at zero

I ε ≥ 0 is the strength of the pinning interaction

The general wetting model

The field ϕ = ϕi1≤i≤N in the interacting case:

Pwε,N

(dϕ1 , . . . , dϕN

e−HN(ϕ)

Zwε,N

N∏i=1

i + ε · δ0(dϕi ))

I dϕ+i is the Lebesgue measure on [0,∞)

I HN(ϕ) describes the structure of the chain (to be specified)

I Zwε,N is the normalization constant (partition function)

The general pinning model

Analogous to the wetting case but without repulsion: dϕ+i → dϕi

Ppε,N

(dϕ1 , . . . , dϕN

e−HN(ϕ)

Zpε,N

N∏i=1

(dϕi + ε · δ0(dϕi )

I dϕi is the Lebesgue measure on RI HN(ϕ) is the same Hamiltonian (to be specified)

I Zpε,N is the normalization constant (partition function)

The general pinning model

Analogous to the wetting case but without repulsion: dϕ+i → dϕi

Ppε,N

(dϕ1 , . . . , dϕN

e−HN(ϕ)

Zpε,N

N∏i=1

(dϕi + ε · δ0(dϕi )

I dϕi is the Lebesgue measure on RI HN(ϕ) is the same Hamiltonian (to be specified)

I Zpε,N is the normalization constant (partition function)

Pinning VS Wetting (+ boundary conditions)(1+1)–DIMENSIONAL FIELDS WITH LAPLACIAN INTERACTION 3

pinning model Pp!,N

wetting model Pw!,N

Figure 1. A graphical representation of the pinning model Pp!,N (top)

and of the wetting model Pw!,N (bottom), for N = 25 and " > 0. The

trajectories (n,!n)0!n!N of the field describe the configurations of alinear chain attracted to a defect line, the x–axis. The grey circles representthe pinned sites, i.e. the points in which the chain touches the defect line,which are energetically favored. Note that in the pinning case the chain cancross the defect line without touching it, while this does not happen in thewetting case due to the presence of a wall, i.e. of a constraint for the chainto stay non-negative: the repulsion e!ect of entropic nature that arises isresponsible for the di!erent critical behavior of the models.

1.2. The free energy and the main results. A convenient way to define localizationfor our models is by looking at the Laplace asymptotic behavior of the partition functionZa

!,N as N "#. More precisely, for a $ p,w we define the free energy fa(") by

fa(") := limN"#

faN (") , fa

N (") :=1N

logZa!,N , (1.8)

where the existence of this limit (that will follow as a by-product of our approach) can beproven with a standard super-additivity argument. The basic observation is that the freeenergy is non-negative. In fact, setting "p := [0,#) and "w := R, we have %N $ N

Za!,N =

"!H[$1,N+1](!)# N$1$

"" #0(d!i) + d!i 1("i%!a)

"!H[$1,N+1](!)# N$1$

d!i 1("i%!a) = Za0,N & c1

(1+1)–DIMENSIONAL FIELDS WITH LAPLACIAN INTERACTION 3

pinning model Pp!,N

wetting model Pw!,N

fa(") := limN"#

faN (") , fa

N (") :=1N

logZa!,N , (1.8)

Za!,N =

"!H[$1,N+1](!)# N$1$

"" #0(d!i) + d!i 1("i%!a)

"!H[$1,N+1](!)# N$1$

d!i 1("i%!a) = Za0,N & c1

Pinning VS Wetting (+ boundary conditions)(1+1)–DIMENSIONAL FIELDS WITH LAPLACIAN INTERACTION 3

pinning model Pp!,N

wetting model Pw!,N

fa(") := limN"#

faN (") , fa

N (") :=1N

logZa!,N , (1.8)

Za!,N =

"!H[$1,N+1](!)# N$1$

"" #0(d!i) + d!i 1("i%!a)

"!H[$1,N+1](!)# N$1$

d!i 1("i%!a) = Za0,N & c1

(1+1)–DIMENSIONAL FIELDS WITH LAPLACIAN INTERACTION 3

pinning model Pp!,N

wetting model Pw!,N

fa(") := limN"#

faN (") , fa

N (") :=1N

logZa!,N , (1.8)

Za!,N =

"!H[$1,N+1](!)# N$1$

"" #0(d!i) + d!i 1("i%!a)

"!H[$1,N+1](!)# N$1$

d!i 1("i%!a) = Za0,N & c1

Some questions

Once HN(ϕ) is chosen and ε ≥ 0 is fixed:

I What are the properties of Ppε,N and Pw

ε,N for large N ?

I Is the field localized at ϕ = 0 or delocalized in ϕ 6= 0?(How to define localization and delocalization?)

I Does the answer depend on ε ≥ 0 (phase transition) and/oron the choice of HN(ϕ)?

How to choose HN(ϕ) ?

Some questions

ε,N for large N ?

Some questions

ε,N for large N ?

Some questions

ε,N for large N ?

The choice of HN(ϕ)

The simplest choice is the gradient case:

HN(ϕ) :=N∑

V(∇ϕi

), ∇ϕi := ϕi − ϕi−1 ,

V (·) : R→ R ∪ +∞ :

e−V (x) dx < +∞ ( = 1 ) .

+ regularity (see later). Gaussian case: V (x) ∝ x2

I Pp0,N is (the bridge of) a random walk of step law e−V (x)

I Pw0,N is (the bridge of) a random walk conditioned to stay ≥ 0

[Isozaki, Yoshida SPA 01] [Deuschel, Giacomin, Zambotti PTRF 05]

[Caravenna, Giacomin, Zambotti EJP 06]

HN(ϕ) :=N∑

V(∇ϕi

), ∇ϕi := ϕi − ϕi−1 ,

V (·) : R→ R ∪ +∞ :

e−V (x) dx < +∞ ( = 1 ) .

HN(ϕ) :=N∑

V(∇ϕi

), ∇ϕi := ϕi − ϕi−1 ,

V (·) : R→ R ∪ +∞ :

e−V (x) dx < +∞ ( = 1 ) .

HN(ϕ) :=N∑

V(∇ϕi

), ∇ϕi := ϕi − ϕi−1 ,

V (·) : R→ R ∪ +∞ :

e−V (x) dx < +∞ ( = 1 ) .

We rather consider the Laplacian case:

HN(ϕ) :=N∑

V(∆ϕi

+ V(∇ϕi

∆ϕi := ∇ϕi+1 −∇ϕi = ϕi+1 + ϕi−1 − 2ϕi

I Simplest non-nearest neighbor interaction

I Semi-flexible polymers: ∆ favors affine configurations

I ∇ and ∆ together?

Interpretation of the free case ε = 0:

I Pp0,N is (the bridge of) the integral of a random walk

I Pw0,N is further conditioned to stay ≥ 0

HN(ϕ) :=N∑

V(∆ϕi

+ V(∇ϕi

∆ϕi := ∇ϕi+1 −∇ϕi = ϕi+1 + ϕi−1 − 2ϕi

HN(ϕ) :=N∑

V(∆ϕi

(∇ϕi

)∆ϕi := ∇ϕi+1 −∇ϕi = ϕi+1 + ϕi−1 − 2ϕi

HN(ϕ) :=N∑

V(∆ϕi

+ V(∇ϕi

∆ϕi := ∇ϕi+1 −∇ϕi = ϕi+1 + ϕi−1 − 2ϕi

Laplacian interaction in (d + 1)-dimension

Fields ϕ : 1, . . . ,Nd → R with Laplacian interaction for d ≥ 2are models for semiflexible membranes

The problem of entropic repulsion (probability for the field to staynon-negative) has been studied in the Gaussian case:

[Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07]

Henceforth we study Ppε,N and Pw

ε,N with Laplacian interaction andboundary conditions ϕ−1 = ϕ0 = ϕN = ϕN+1 = 0Assumptions on V :∫

Re−V (x) dx = 1 ,

x e−V (x) dx = 0 ,

x2 e−V (x) dx = 1

+ regularity: x 7→ e−V (x) continuous and V (0) < +∞.

Re−V (x) dx = 1 ,

x e−V (x) dx = 0 ,

x2 e−V (x) dx = 1

ε,N with Laplacian interaction andboundary conditions ϕ−1 = ϕ0 = ϕN = ϕN+1 = 0

Assumptions on V :∫R

e−V (x) dx = 1 ,

x e−V (x) dx = 0 ,

x2 e−V (x) dx = 1

Re−V (x) dx = 1 ,

x e−V (x) dx = 0 ,

x2 e−V (x) dx = 1

Outline

The Models Free Energy Path Results Proofs The free energy

How to define localization and delocalization?

Recall the partition function: (zero boundary conditions)

Zaε,N =

∫Ωa

e−HN(ϕ)N−1∏i=1

(dϕi + ε δ0(dϕi )

)where a ∈ p,w and Ωp

N = RN−1 while ΩwN = [0,∞)N−1

Free Energy

Fa(ε) := limN→∞

Nlog Za

ε,N (super-additivity)

Basic observation: Fa(ε) ≥ Fa(0) = 0 for all ε ≥ 0 and a ∈ p,w

Zaε,N ≥ Za

0,N ≈ N−c (c > 0)

Zaε,N =

∫Ωa

Free Energy

Nlog Za

Zaε,N ≥ Za

0,N ≈ N−c (c > 0)

Zaε,N =

∫Ωa

Free Energy

Nlog Za

Zaε,N ≥ Za

0,N ≈ N−c (c > 0)

Localization and delocalization

Definition

The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0.

localized ⇐⇒ ε > εac := supε ≥ 0 : Fa(ε) = 0 ∈ [0,∞]

Setting `N := #

i ≤ N : ϕi = 0

we have:

I if ε > εac then`NN→ Da(ε) > 0 in Pa

ε,N –probability

I if ε < εac then`NN→ 0 in Pa

ε,N –probability (delocalization)

I if ε = εac ? Depends on the model:

I if Fa(εac + h) = o(h) [> 1st order trans.] ε = εac is delocalized

I if Fa(εac + h) ≥ C h [1st order trans.] ε = εac may be localized(phase coexistence, dependence of boundary conditions)

Definition

Setting `N := #

i ≤ N : ϕi = 0

we have:

ε,N –probability

Definition

Setting `N := #

i ≤ N : ϕi = 0

we have:

ε,N –probability

Definition

Setting `N := #

i ≤ N : ϕi = 0

we have:

ε,N –probability

Definition

Setting `N := #

i ≤ N : ϕi = 0

we have:

ε,N –probability

Definition

Setting `N := #

i ≤ N : ϕi = 0

we have:

ε,N –probability

Definition

Setting `N := #

i ≤ N : ϕi = 0

we have:

ε,N –probability

The Models Free Energy Path Results Proofs The phase transition

The phase transition

Theorem ([CD1])

Both Ppε,N and Pw

ε,N undergo a non-trivial phase transition:

0 < εpc < εwc < ∞

and Fa(ε) is analytic on [0, εac) ∪ (εac ,∞). (variational formula)

I In the pinning model the transition is exactly of 2nd order:

log 1h

≤ Fp(εpc + h) ≤ o(h)

I In the wetting model the transition is of 1st order:

Fw(εwc + h) ∼ C2 h , `N ∼ DN (D > 0)

Theorem ([CD1])

Both Ppε,N and Pw

0 < εpc < εwc < ∞

log 1h

Fw(εwc + h) ∼ C2 h , `N ∼ DN (D > 0)

Theorem ([CD1])

Both Ppε,N and Pw

0 < εpc < εwc < ∞

log 1h

Fw(εwc + h) ∼ C2 h , `N ∼ DN (D > 0)

The gradient case

Differences in the gradient case

I the transition is non-trivial only in the wetting model:

εp,∇c = 0 , 0 < εw,∇c < ∞

I the transition is of 2nd order:

Fp(εp,∇c + h

) ∼ Cp h2 , Fw(εw,∇c + h

) ∼ Cw h2

INSERT SHORTER TITLE (IF TITLE IS TOO LONG)! 3

BestJ-DConcerning the repulsion, I wonder whether a simple FKG argument would work:

Consider the gaussian field Ax, x = 0, 1, 2, ... where Ax = S1 + ...Sx and Sx =X1 + ...Xx Xi are i.i.d. gaussian. Note that cov(Ax, Ay) ! 0. That is the field Axsatisfies the FKG property. Let P+

N = P (·|AN!1 ! 0, AN ! 0). Now assume that wecan show that P+

N is also FKG. Let !+N = Sx ! 0, x = 1, 2, ..., N, and for ! > 0,

!!N (!) = SN!1, SN " !. By FKG

P+N (!+

N # !!N(!)) " P+N (!+

N )P+N (!!N (!))

P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)

P (0 " SN!1 " !, 0 " SN " !)" P (!+

P (SN!1 ! 0, SN ! 0)" 4P (!+

since again by FKG

P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1

Now, letting !$ 0 we are done.In order to prove FKG for P+

N , we can use the fact that a ”product conditioning”does not destroy FKG, but it is a bit subtle, so I have to think about it !

BestJ-D

""pc"p,"

Figure 1. Write caption.

References

[1] W. Feller (1968) An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed.,John Wiley and Sons, New York

Dipartimento di Matematica Pura e Applicata, Universita di Padova, via Trieste63, 35121 Padova, Italy

E-mail address: [email protected]

4 FRANCESCO CARAVENNA

!!wc!w,!

The gradient case

εp,∇c = 0 , 0 < εw,∇c < ∞I the transition is of 2nd order:

Fp(εp,∇c + h

) ∼ Cp h2 , Fw(εw,∇c + h

) ∼ Cw h2

!!N (!) = SN!1, SN " !. By FKG

P+N (!+

N # !!N(!)) " P+N (!+

N )P+N (!!N (!))

P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)

P (0 " SN!1 " !, 0 " SN " !)" P (!+

P (SN!1 ! 0, SN ! 0)" 4P (!+

since again by FKG

P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1

BestJ-D

""pc"p,"

References

!!wc!w,!

The gradient case

εp,∇c = 0 , 0 < εw,∇c < ∞I the transition is of 2nd order:

Fp(εp,∇c + h

) ∼ Cp h2 , Fw(εw,∇c + h

) ∼ Cw h2

!!N (!) = SN!1, SN " !. By FKG

P+N (!+

N # !!N(!)) " P+N (!+

N )P+N (!!N (!))

P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)

P (0 " SN!1 " !, 0 " SN " !)" P (!+

P (SN!1 ! 0, SN ! 0)" 4P (!+

since again by FKG

P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1

BestJ-D

""pc"p,"

References

!!wc!w,!

Figure 2. Write caption.Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 17 / 34

The Models Free Energy Path Results Proofs The disordered case

A look at the disordered case

Disordered version of our model: (dϕpi = dϕi and dϕw

i = dϕ+i )

Paε,β,ω,N

(dϕ1 , . . . , dϕN

e−HN(ϕ)

Zaε,β,ω,N

N∏i=1

i + ε eβωi δ0(dϕi ))

where β ≥ 0 and ωii∈N are IID N (0, 1) (law P indep. Pa).

Quenched free energy

Fa(ε, β)

:= limN→∞

Nlog Za

ε,β,ω,N ≥ 0

exists P(dω)–a.s. and does not depend on ω (self-averaging)

Localization: Fa(ε, β) > 0 ⇐⇒ ε > εac(β) (critical line)

i = dϕ+i )

Paε,β,ω,N

(dϕ1 , . . . , dϕN

e−HN(ϕ)

Zaε,β,ω,N

N∏i=1

Fa(ε, β)

:= limN→∞

Nlog Za

ε,β,ω,N ≥ 0

i = dϕ+i )

Paε,β,ω,N

(dϕ1 , . . . , dϕN

e−HN(ϕ)

Zaε,β,ω,N

N∏i=1

Fa(ε, β)

:= limN→∞

Nlog Za

ε,β,ω,N ≥ 0

Smoothing effect of disorder

What is the behavior of εac(β) for small β ? (εac = εac(0))

What is the regularity of the transition in the disordered case?

Theorem ([Giacomin and Toninelli, CMP 06])

Both in the ∇ and ∆ case, both for a = p and for a = w:for every β > 0 there exists Cβ > 0 such that

Fa(εac(β) + h

) ≤ Cβ h2

When disorder is present the transition is at least of 2nd orderSharp contrast with homogeneous case cf.

Very general proof: rare-stretches in ω (Large Deviations)

Fa(εac(β) + h

) ≤ Cβ h2

Fa(εac(β) + h

) ≤ Cβ h2

Outline

Some deeper questions

We have established the existence of a phase transition:

`N = o(N) if ε < εac VS `N ∼ D · N if ε > εac

Can we say something more precise? Typical paths of Paε,N ?

Yes in the pinning case and under additional assumptions on V (·):

I symmetry: V (x) = V (−x) for every x ∈ RI uniform strict convexity: ∃γ > 0 s. t. V (x)− γ x2

2 is convex

I regularity: x 7→ e−V (x) is continuous and V (0) <∞∫R

e−V (x) dx = 1

x2 e−V (x) dx = 1

Let us set MN := max1≤i≤N

∣∣ϕi

∣∣

2 is convex

e−V (x) dx = 1

x2 e−V (x) dx = 1

∣∣ϕi

∣∣

2 is convex

e−V (x) dx = 1

x2 e−V (x) dx = 1

∣∣ϕi

∣∣Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 21 / 34

The Models Free Energy Path Results Proofs Path results

A closer look at the typical paths [CD2]

I delocalized regime ε < εpc : `N = O(1) and MN ≈ N3/2 :

limK→∞

lim infN→∞

Ppε,N

(ϕi 6= 0 for i ∈ K ,N − K) = 1

limK→∞

lim infN→∞

Ppε,N

KN3/2 ≤ MN ≤ K N3/2

I localized regime ε > εpc : MN = O((log N)2

limK→∞

lim infN→∞

Ppε,N

(MN ≤ K (log N)2

I critical regime ε = εpc : MN ≈ N3/2

(log N)c and `N ≈ Nlog N :

limK→∞

lim infN→∞

Ppεpc ,N

(log N)3/2≤ MN ≤ K

limK→∞

lim infN→∞

Ppε,N

(ϕi 6= 0 for i ∈ K ,N − K) = 1

limK→∞

lim infN→∞

Ppε,N

KN3/2 ≤ MN ≤ K N3/2

limK→∞

lim infN→∞

Ppε,N

(MN ≤ K (log N)2

limK→∞

lim infN→∞

Ppεpc ,N

(log N)3/2≤ MN ≤ K

limK→∞

lim infN→∞

Ppε,N

(ϕi 6= 0 for i ∈ K ,N − K) = 1

limK→∞

lim infN→∞

Ppε,N

KN3/2 ≤ MN ≤ K N3/2

limK→∞

lim infN→∞

Ppε,N

(MN ≤ K (log N)2

limK→∞

lim infN→∞

Ppεpc ,N

(log N)3/2≤ MN ≤ K

Scaling Limits

We rescale and interpolate linearly the field: for t ∈ [0, 1]

ϕN(t) :=ϕbNtcN3/2

+ (Nt − bNtc) ϕbNtc+1 − ϕbNtcN3/2

Let Btt∈[0,1] standard BM, It :=∫ t

0 Bs ds integrated BM(Bt , It)

t∈[0,1]

(Bt , It)

t∈[0,1]cond. on (B1, I1) = (0, 0)

Theorem (Scaling Limits [CD2])

The rescaled field ϕN(t)t∈[0,1] under Ppε,N converges in

distribution on C ([0, 1]) as N →∞, for every ε ≥ 0. The limit is

I If ε < εpc , the law of Itt∈[0,1]

I If ε = εpc or ε > εpc , the law concentrated on f (t) ≡ 0

Scaling Limits

ϕN(t) :=ϕbNtcN3/2

t∈[0,1]

(Bt , It)

t∈[0,1]cond. on (B1, I1) = (0, 0)

Scaling Limits

ϕN(t) :=ϕbNtcN3/2

t∈[0,1]

(Bt , It)

t∈[0,1]cond. on (B1, I1) = (0, 0)

The Models Free Energy Path Results Proofs Refined critical scaling limit

The critical regime

For ε = εpc the field has very large fluctuations (≈ N3/2

(log N)c ).Can we extract a non-trivial scaling limit?

Not in C ([0, 1]) or D([0, 1]):

i ∈ 1, . . . ,N : ϕi = 0

becomes dense in [0, 1]

!!N (!) = SN!1, SN " !. By FKG

P+N (!+

N # !!N(!)) " P+N (!+

N )P+N (!!N (!))

P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)

P (0 " SN!1 " !, 0 " SN " !)" P (!+

P (SN!1 ! 0, SN ! 0)" 4P (!+

since again by FKG

P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1

BestJ-D

(log N)3/2

References

The critical regime

Not in C ([0, 1]) or D([0, 1]):

i ∈ 1, . . . ,N : ϕi = 0

!!N (!) = SN!1, SN " !. By FKG

P+N (!+

N # !!N(!)) " P+N (!+

N )P+N (!!N (!))

P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)

P (0 " SN!1 " !, 0 " SN " !)" P (!+

P (SN!1 ! 0, SN ! 0)" 4P (!+

since again by FKG

P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1

BestJ-D

(log N)3/2

References

The critical regime

Not in C ([0, 1]) or D([0, 1]):

i ∈ 1, . . . ,N : ϕi = 0

!!N (!) = SN!1, SN " !. By FKG

P+N (!+

N # !!N(!)) " P+N (!+

N )P+N (!!N (!))

P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)

P (0 " SN!1 " !, 0 " SN " !)" P (!+

P (SN!1 ! 0, SN ! 0)" 4P (!+

since again by FKG

P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1

BestJ-D

(log N)3/2

References

The critical regime

Alternative idea: look at the field in a distributional sense

:=(log N)5/2

N3/2ϕbNtc dt = ϕN(t) dt

µN(·) is a (random) finite signed measure on [0, 1]

Let Ltt∈[0,1] be the stable symmetric Levy process of index 25

0 drift 0 Brownian component Π(dx) =c

|x |7/5dx

The paths of L are a.s. of bounded variation, hence we set

dL((a, b)

):= Lb − La

Theorem ([CD2])

The random signed measure µN under Ppεpc ,N

converges in

distribution as N →∞ toward dL .

The critical regime

:=(log N)5/2

|x |7/5dx

dL((a, b)

):= Lb − La

Theorem ([CD2])

converges in

The critical regime

:=(log N)5/2

|x |7/5dx

dL((a, b)

):= Lb − La

Theorem ([CD2])

converges in

The critical regime

:=(log N)5/2

|x |7/5dx

dL((a, b)

):= Lb − La

Theorem ([CD2])

converges in

The critical regime6 FRANCESCO CARAVENNA AND JEAN–DOMINIQUE DEUSCHEL

!!N (t)

# O$log N

Figure 1. A graphical representation of Theorem 1.6. For large N , theexcursions of the rescaled field under the critical law P!c,N contribute to themeasure µN (dt), see (1.18), approximately like Dirac masses, with intensitygiven by their (signed) area. The width and height of the relevant excursionsare of order (1/ log N) and log N respectively. We warn the reader that thex- and y-axis in the picture have di!erent units of length, and that the fieldcan actually cross the x-axis without touching it (though this feature hasnot been evidenced in the picture for simplicity).

We look at µN under the critical law P!c,N as a random element of M([0, 1]), the space ofall finite signed Borel measures on the interval [0, 1], that we equip with the topology ofvague convergence and with the corresponding Borel "–field (#n ! # vaguely if and onlyif

&fd#n !

&fd# for all bounded and continuous functions f : [0, 1] ! R). Our goal is

to show that the sequence µNN has a non-trivial limit in distribution on M([0, 1]).To describe the limit, let Ltt!0 denote the stable symmetric Levy process of index 2/5

(a standard version with cadlag paths). More explicitly, Ltt!0 is a Levy process with zerodrift, zero Brownian component and with Levy measure given by "(dx) = cL |x|"7/5dx,where the positive constant cL is defined explicitly in equation (6.25). Since the index isless than 1, the paths of L are a.s. of bounded variation, cf. [2], hence we can define pathby path the (random) finite signed measure dL in the Steltjes sense:

dL$(a, b]

%:= Lb " La .

We stress that dL is a.s. a purely atomic measure, i.e., a sum of Dirac masses (for moredetails and for an explicit construction of dL, see Remark 1.7 below).

We are now ready to state our main result (see Figure 1 for a graphical description).

Theorem 1.6. The random signed measure µN under P!c,N converges in distribution onM([0, 1]) as N ! # toward the the random signed measure dL.

Outline

The Models Free Energy Path Results Proofs Integrated random walk

A random walk viewpoint (ε = 0)

Let Xii∈N be IID random variables with law

P(Xi ∈ dx) := e−V (x) dx E(Xi ) = 0 Var(Xi ) = 1

Random walk: Yn := X1 + . . .+ Xn

Integrated random walk:

Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn

The free case ε = 0

I The field ϕi1≤i≤N under Pp0,N is distributed like Zi1≤i≤N

conditionally on (YN ,ZN) = (0, 0). (ZN/2 ≈ N3/2)

I Under Pw0,N the same, under the further conditioning

Z1 ≥ 0, . . . ,ZN ≥ 0

Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn

Z1 ≥ 0, . . . ,ZN ≥ 0

Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn

Z1 ≥ 0, . . . ,ZN ≥ 0

Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn

Z1 ≥ 0, . . . ,ZN ≥ 0

Excursions and contact set

What happens for ε > 0? The same holds for the excursions.

Contact set: τ :=

i ∈ N : ϕi = 0

= τii≥0

Excursions: ei (k)k := ϕτi−1+k0≤k≤τi−τi−1for i ∈ N

Auxiliary chain: Ji := ϕτi−1 for i ∈ N

Conditionally on τ and J, the excursions ei (·)i∈N under Paε,N are

independent and distributed:

I for a = p like bridges of the process ZiiI for a = w like bridges of the process Zii conditioned to stay

non-negative

Once we know τ, J, the whole field ϕii is reconstructed bypasting independent excursions from Zii (cond. to stay ≥ 0)

Contact set: τ :=

i ∈ N : ϕi = 0

= τii≥0

non-negative

Contact set: τ :=

i ∈ N : ϕi = 0

= τii≥0

non-negative

Contact set: τ :=

i ∈ N : ϕi = 0

= τii≥0

non-negative

Contact set: τ :=

i ∈ N : ϕi = 0

= τii≥0

non-negative

Contact set: τ :=

i ∈ N : ϕi = 0

= τii≥0

I for a = p like bridges of the process Zii

I for a = w like bridges of the process Zii conditioned to staynon-negative

Contact set: τ :=

i ∈ N : ϕi = 0

= τii≥0

non-negative

Contact set: τ :=

i ∈ N : ϕi = 0

= τii≥0

non-negative

The law of the excursions

Pinning case: good control (Donsker’s inv. pr. + LLT)Z〈Nt〉N3/2

t∈[0,1]

condit. on (YN ,ZN) = (0, 0) =⇒

t∈[0,1]

Wetting case: several open issues

I Integrated BM conditioned to stay non-negative is studied,but not its bridge

I Invariance principle seems in any case very difficult

I Entropic repulsion:

P(Z1 ≥ 0 , . . . , ZN ≥ 0

) ≈P(Z1 ≥ 0 , . . . , ZN ≥ 0

∣∣YN = 0,ZN = 0) ≈

t∈[0,1]

condit. on (YN ,ZN) = (0, 0) =⇒

t∈[0,1]

P(Z1 ≥ 0 , . . . , ZN ≥ 0

) ≈P(Z1 ≥ 0 , . . . , ZN ≥ 0

∣∣YN = 0,ZN = 0) ≈

t∈[0,1]

condit. on (YN ,ZN) = (0, 0) =⇒

t∈[0,1]

P(Z1 ≥ 0 , . . . , ZN ≥ 0

) ≈P(Z1 ≥ 0 , . . . , ZN ≥ 0

∣∣YN = 0,ZN = 0) ≈

t∈[0,1]

condit. on (YN ,ZN) = (0, 0) =⇒

t∈[0,1]

P(Z1 ≥ 0 , . . . , ZN ≥ 0

) ≈ ?

P(Z1 ≥ 0 , . . . , ZN ≥ 0

∣∣YN = 0,ZN = 0) ≈ ?

t∈[0,1]

condit. on (YN ,ZN) = (0, 0) =⇒

t∈[0,1]

P(Z1 ≥ 0 , . . . , ZN ≥ 0

) ≈ N−1/4 [Sinai (SRW)]

P(Z1 ≥ 0 , . . . , ZN ≥ 0

∣∣YN = 0,ZN = 0) ≈ N−1/2 [conj.]

The Models Free Energy Path Results Proofs Markov renewal theory

Markov renewal processes

Given a (sub-)probability kernel Kx ,dy (n):∫y∈R

∑n∈N

Kx ,dy (n) = c ≤ 1 , ∀x ∈ R

we build the Markov renewal process τ with modulating chain J:

P(τi+1 − τi = n , Ji+1 ∈ dy

∣∣∣ Ji = x)

:= Kx ,dy (n)

The law of (τ, J) conditionally on N,N + 1 ⊆ τ is

P (τi = ti , Ji ∈ dyi

∣∣ N,N + 1 ⊆ τ) =1

Kyi−1,dyi(ti − ti−1)

with CN = P(N,N + 1 ⊆ τ).

Given a (sub-)probability kernel Kx ,dy (n):∫y∈R

∑n∈N

Kx ,dy (n) = c ≤ 1 , ∀x ∈ R

we build the Markov renewal process τ with modulating chain J:

P(τi+1 − τi = n , Ji+1 ∈ dy

∣∣∣ Ji = x)

:= Kx ,dy (n)

The law of (τ, J) conditionally on N,N + 1 ⊆ τ is

P (τi = ti , Ji ∈ dyi

∣∣ N,N + 1 ⊆ τ) =1

Kyi−1,dyi(ti − ti−1)

with CN = P(N,N + 1 ⊆ τ).

The law of the contact set

Consider the following kernels: for n ∈ N and x , y ∈ R

Gpx ,dy (n) := ε

(Zn−1 ∈ dy , Zn ∈ dz

∣∣∣∣z=0

Gwx ,dy (n) := ε

(Zi ≥ 0 , i ≤ n , Zn−1 ∈ dy , Zn ∈ dz

∣∣∣∣z=0

The law of (τ, J) is given by

Paε,N

(τi = ti , Ji ∈ dyi , i ≤ k

Zaε,N

k∏i=1

Gayi−1,dyi

(ti − ti−1)

Reminds of Markov renewal theory

Gpx ,dy (n) := ε

∣∣∣∣z=0

Gwx ,dy (n) := ε

(Zi ≥ 0 , i ≤ n , Zn−1 ∈ dy , Zn ∈ dz

∣∣∣∣z=0

Paε,N

Zaε,N

k∏i=1

Gayi−1,dyi

(ti − ti−1)

Gpx ,dy (n) := ε

∣∣∣∣z=0

Gwx ,dy (n) := ε

(Zi ≥ 0 , i ≤ n , Zn−1 ∈ dy , Zn ∈ dz

∣∣∣∣z=0

Paε,N

Zaε,N

k∏i=1

Gayi−1,dyi

(ti − ti−1)

We exploit the invariance properties: for every F, v(y)

Paε,N

Zaε,N

k∏i=1

Gayi−1,dyi

(ti − ti−1) e−F(ti−ti−1) v(yi )

v(yi−1)

If we determine F, v(·) such that

Kx ,dy (n) := Gax ,dy (n) e−F·n

is a probability kernel, we have the crucial relation

Paε,N

(τi = ti , Ji ∈ dyi

)= P(τi = ti , Ji ∈ dyi

∣∣ N,N + 1 ⊆ τ)

We exploit the invariance properties: for every F, v(y)

Paε,N

Zaε,N

k∏i=1

Gayi−1,dyi

(ti − ti−1) e−F(ti−ti−1) v(yi )

v(yi−1)

If we determine F, v(·) such that

Kx ,dy (n) := Gax ,dy (n) e−F·n

is a probability kernel, we have the crucial relation

Paε,N

(τi = ti , Ji ∈ dyi

)= P(τi = ti , Ji ∈ dyi

∣∣ N,N + 1 ⊆ τ)Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 33 / 34

A Perron-Frobenius problem

It turns out that:

I F is the solution of the equation

spectral radius of

(∑n∈N

Gax ,dy (n) e−F·n

when such a solution exists and F = 0 otherwise.

I In fact F = Fa(ε) is the free energy

I v(·) is the principal eigenfunction:∫y∈R

(∑n∈N

v(y) = v(x)

It turns out that:

spectral radius of

(∑n∈N

v(y) = v(x)

It turns out that:

spectral radius of

(∑n∈N

v(y) = v(x)

pinning and wetting transition for (1+1)-dimensional...

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