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Activated Random Walks on Z d Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar – July 2020

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  • Activated Random Walks on Zd

    Lectures 9 and 10

    Leonardo T. Rolla

    THU-PKU-BNU joint probability webinar – July 2020

  • Tentative plan

  • Tentative plan

    Recall of previous lectures

    Particle-wise construction is well-defined [§11.3]

    Uniqueness of the critical density [§8]

    Weak and strong stabilization [§7]

  • Recall

  • Recall

    Dynamics and phase space

    Odometer and toppling procedures

    Counting arguments

    Exploring instructions in advance

    Coarse-grained flow between blocks

    The particle-wise construction and applications

  • Phase space [§1.5]


    d = 1 directed d = 1 biased d = 1 unbiased

    d = 2 unbiasedd ≥ 3 unbiasedd ≥ 2 biased



    scaling limit



















  • Odometer and Abelian property [§2.2]All deterministic: finite sequences of topplings α, β

    mα(x) := #times x appears in α

    mV,η := supβ⊆V legal


    mV,η 6 mα if α stabilizes η in V

    mV,η ↑ mη

    Now make it random

  • Counting arguments [§3]

















    · · · · · ·

    −L+ 1 −L+ 2 · · ·


  • Counting arguments [§3]


    (Taggi; R, Tournier)

  • Exploring instructions in advance [§4]

    (R, Sidoravicius)

  • Coarse-grained flow between blocks [§5]

    Analyze the odometer m1, . . . ,mn at the buffers

    Mass Balance Equations and Single-Block Dynamics

    (Basu, Ganguly, Hoffman)

  • The particle-wise construction [§10.1]

    Labeled particles

    Constructed from η0, CTRW, Clocks

    Defined through a limit (see below)

    Fixation equivalence: Sites fixate ⇔ Particles fixate

    Conservation: If fixate, E[start at 0] = E[settle at 0]

    Corollary: ζc 6 1

  • Averaged condition for activity [§10.2]lim supn


    > 0 =⇒ Activity


    m (R, Tournier)

  • Fixation equivalence [§10.4]Theorem 10.7. Sites fixate ⇔ Particles fixate

    Idea: extra randomness so as to spread out the effect

    of non-fixating particles and control variance.

    Implementation: for each particle that stays active, tag

    at random one of the n first sites visited after time t.

    (Amir, Gurel-Gurevich)

  • Resampling [§10.3]Theorem 10.4. For i.i.d. random initial configurations

    with average ζ = 1, the system a.s. stays active.

    Take λ =∞, change rates to particle-hole model.

    Suppose finitely many particles visit 0.

    Resample η0 on the sites where they may start, so

    wpp site 0 is never visited. Conclude that ζ < 1.

    (Cabezas, R, Sidoravicius)

  • §11.3The particle-wise construction

    is well-defined

  • Construction [§10.1]

    For a triple (η0,X,P), we we say that(η0,X,P) 7→ (η0,Y ,γ) is well-defined if:

    (i) for each x, y ∈ Zd, j ∈ N and t > 0, both(Y x,js )s∈[0,t] and (γ

    x,js )s∈[0,t] are the same in the

    systems (η0 · 1Byn ,X,P) for all but finitely many n;

    (ii) the limit (η0,Y ,γ) does not depend on y.

  • Statement

    Theorem 10.6. If supx E|η0(x)|

  • Overview of the proof

    ` For arbitrary fixed Vn ↑ Zd there is an a.s. limit.

    Add particles one by one, updating the whole evolution

    ` Life of each particle is well-defined through some limit

    Main step: ∀x, T , the number of particle additions thataffect site x by time T has finite expectation

  • Tracking differences

  • Tracking differences

  • Tracking differences

  • Tracking differences

  • Tracking differences

  • Tracking differences

  • Tracking differences

  • Tracking differences

  • Tracking differences

  • Tracking differences

  • Tracking differences

    Dominate by a supercritical branching process.

    E[green] = e(2+λ)t

    Re-index sums etc, Borel-Cantelli...

  • §8Uniqueness of the critical density

  • Uniqueness of the critical density

    Theorem 2.13. Given the dimension d, sleep rate λ,

    and jump distribution p(·), there is a number ζc suchthat, for every translation-ergodic distribution ν

    supported on (N0)Zd

    with average density ζ, the ARW

    dynamics satisfies

    Pν(system stays active) =

    0, ζ < ζc,1, ζ > ζc.(R, Sidoravicius, Zindy)

  • Equivalent statement

    Theorem 8.1. Let d, λ and p(·) be given. Let ν1 andν2 be two spatially ergodic distributions on (N0)Z


    with respective densities ζ1 < ζ2. If the ARW system is

    a.s. fixating with initial state ν2, then it is also a.s.

    fixating with initial state ν1.

  • The more general version

    Open Problem. Suppose ν is a translation-ergodic

    active state (active means ν is supported on

    (Ns)Zd \ {0, s}Zd) with density ζ > ζc. Show that the

    ARW with initial state ν a.s. stays active.

  • Idea of the proof

    - Embedding the initial configuration into another one

    with higher density (decoupling)

    - Stabilization of the embedded configuration

    - Stabilization of the original configuration

    Remark. Not a sequential procedure like previous ones

  • Decoupling

    Sample η0 ∼ ν1, ξ0 ∼ ν2 and I independently → ω.

    Assume wlog ν1 or ν2 mixing, hence ω ergodic.

    ` A doubly-infinite procedure which is a factor of ω.

    Let A0 = {x : η0(x) > ξ0(x)}. Topple every site in A0.Result η1 is insensitive to the order, hence a factor.

    Repeat for η0, η1, η2, . . . . Limit η∞ = η′0 exists.

    Each site is toppled finitely often, otherwise ζ1 > ζ2.

  • Stabilization of the larger configuration

    Delete the instructions used in the previous stage.

    Zero out odometer.

    Conditioning on the outcome of the first step, the

    remaining instructions are again i.i.d. with the correct


    Stabilize ξ0. Odometer mξ0(x) < +∞ by assumption.

  • Stabilization of the original configuration

    Since η′0 6 ξ0, we also have mη′0(x) < +∞.

    Two stages: embedding and then stabilizing.

    Some topplings in the first stage were not legal

    (because we made forced sleepy particles to wake up

    and jump).

    Hence, the sum of the (locally finite) odometers

    obtained in these two stages is an upper bound for the

    odometer of η0.

  • §7Weak and strong stabilization

  • Results

    Theorem 7.1. For any jump distribution in any

    dimension, ζc > λ1+λ .

    Theorem 7.2. If d > 2, then ζc < 1 for every λ

  • Weak and strong stabilization

    We say that 0 is w-stable if η(0) 6 1, and we say that0 is s-stable if η(0) = 0. Otherwise we say that 0 is

    w-unstable or s-unstable. For y 6= 0 we say that y isstable, w-stable, and s-stable if η(y) 6 s.


    mwV,η 6 mV,η 6 msV,η.

    ηwV and η′V : configuration after (weakly) stabilizing

  • Proof of Theorem 7.1

    Using Abelian Property, one way to stabilize η0 on Bn

    is to first weakly stabilize it and then stabilize it.

    ` mV,η0(0) > 1 =⇒ ηwV (0) = 1

    ` P(η′V (0) = s

    )> λ1+λ P

    (ηwV (0) = 1

    )Hence, P

    (η′V (0) = s

    )> λ1+λ P

    (mV,η0(0) > 1

    )Using monotonicity and amenability... non-fixation

    implies ζ > λ1+λ .

  • Jump odometer and extra particles

    Define the “jump odometer” m̄V,η by counting only the

    number of jump instructions performed at each site

    when η is stabilized in V .

    Define m̄sV,η and m̄wV,η similarly.

    Let η+ = η + δ0 denote the result of adding an active

    particle at 0 to a configuration η.

  • Strong − weak = extra particle

    Lemma 7.5. We have m̄sV,η = m̄wV,η+ .

    Proof. A sequence of topplings β is w-legal for η+ if

    and only if it is s-legal for η.

  • Getting rid of the extra particle

    Lemma 7.6. We have E[m̄wV,η+(0)

    ]6 G+E



    Sketch. Force the particle to move.

    Corollary 7.7. E[m̄sV,η(0)− m̄wV,η(0)

    ]6 G.

  • Successive weak stabilizations

    Particle at 0?

    Move on to thenext round



    First round

    Topple 0

    Perform weakstabilization in V






    illegal but acceptable

    there is a single particle at 0

    w-stable: η(0) 6 1stable: η(0) 6 s

    s-stable: η(0) = 0

  • Successive weak stabilizations (cont)

    Let TV and TsV count the number of rounds needed for

    stabilization and strong stabilization to be achieved,

    respectively (weak stabilization is always achieved in the

    first round). From this definition we have

    TV = 1 ⇐⇒ ηwV (0) = 0 ⇐⇒ T sV = 1.

  • Successive weak stabilizations (cont)

    Lemma 7.8. m̄sV,η(0) > m̄wV,η(0) + T sV − 1.

    Corollary 7.9. ETV 6 ET sV 6 1 +G.

  • Proof of the main theorems (overview)

    P(η′V (0) = s



    P(η′(0) = s, TV = n


    P(η′V (0) = s, TV = n) 6 λ1+λ(



    P(η′V (0) = s, TV = n

    ∣∣TV > n) = λ1+λ