Activated Random Walks with bias: activity at low density

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Activated Random Walks with bias:

activity at low density

Laurent TOURNIER

Joint work with Leonardo ROLLA LAGA (Université Paris 13)

Conference “Disordered models of mathematical physics”

Valparaíso, Chile — July 23, 2015

Laurent TOURNIER Activated Random Walks with a bias

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Activated Random Walks (ARW) — Quick presentation

Dynamics:Particles evolve in continuous time onZ^d, and can be either active, instate A: move as (independent) random walks, at rate 1;

passive (sleeping), instate S: do not move.

Two kinds of mutations/interactions happen:

A→Sat rateλ: each particle gets asleep at rateλ (independently);

A+S→2Aimmediately: active particles awake the others on same site.

NB. Mutations A→S are only effective when the particle A is alone

⇒On each site, there is either nothing, one S, or any number of A particles.

Parameters:

jump distributionp(·)onZ^d sleeping rateλ∈(0,∞)

initial configuration of A particles (finite support, or i.i.d. in general).

Behaviors of interest:

fixation: in any finite box, activity vanishes eventually;

non-fixation: in any finite box, activity goes on forever.

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Motivations: 1. Phase transition

Letµdenote the initial density of particles (for i.i.d. initial configuration).

Aphase transitionisexpectedto happen:∃µ_c(λ)∈(0,1)s.t.

forµ<µc(λ), a.s. fixation;

forµ>µ_c(λ), a.s. non-fixation.

Or also: forµ≥1 then a.s. fixation and, forµ<1,∃λ_c(µ)∈(0,∞)s.t.

forλ>λc(µ), a.s. fixation;

forλ<λ_c(µ), a.s. non-fixation.

λ

µ

O 1

Fixation

Non-fixation

Existence ofµ_candλ_cfollow by monotonicity in Diaconis-Fulton coupling.

However, nontrivial bounds are the difficult part→cf. Leo Rolla’s talk for an overview of known results.

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Motivations: 2. Self-Organized Criticality (physics)

Ingredient of a toy example ofself-organized criticality, i.e. a model that displays “critical behavior” (polynomial decay of correlations,...)

spontaneously, without having to tune some parameter:

In a finite box,

Drop a new particle at random,

Stabilize the configuration by running the dynamics inside the box and by freezing particles that exit,

and repeat.

0 500 1000 1500 2000 2500 3000 3500

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

# particles remaining

# particles dropped

,→Dynamics reach a stationary regime which, extended to infinite volume, should satisfy SOC.

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Case of a biased random walk – Main result

Assume the jump distributionp(·)has abias: for the simple random walkX with jump distributionp(·), for some direction`,X_n·`→+∞,a.s..

Forλ>0,v∈R^d\ {0}, ifT_vis the time spent byXin{x∈Z^d:x·v≤0}, let Fv(λ) =Eh 1

(1+λ)^T^v i

=P(a walk killed at rateλ in{x·v≤0}survives forever) NB. Ifv·` >0, then 0<Fv(λ)−→1 asλ →0⁺.

Theorem (Taggi, 2014)

Assume d=1.µ>1−F₁(λ)⇒non-fixation a.s.

Assume d≥2.µFv(λ)>P(η0(0) =0)⇒non-fixation a.s.

Theorem (Rolla-T., 2015)

Assume d≥2.µ>1−Fv(λ)⇒non-fixation a.s.

,→for alld, for allµ<1, non-fixation happens for smallλ.

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Plan

Our proof is based on a mixed use ofparticle-wiseandsite-wiseviewpoints.

1 Definition: particle-wise vs. site-wise

2 Definition: particle fixation vs. site fixation

3 A non-fixation condition(particle-wise+ site-wiseargument)

4 Proof of the result(site-wise argument)

5 Existence of the particle-wise construction.

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Site-wise vs. particle-wise

Thesite-wise viewpointattaches randomness tosites: from finite initial configuration, (“Diaconis-Fulton” construction)

eachsitecontains a random stack of i.i.d.instructions(“jump toy”, or

“sleep”), and a Poisson clock;

when clock rings at a site, apply the top instruction to a particle there;

clock runs at speed proportional to number of particles present at the site (as if each particle reads an instruction at rate 1).

,→we don’t distinguish particles at a site, and getηt(x)∈ {0,S,1,2, . . .}.

Crucial properties:abelianness and monotonicity.

Theparticle-wise viewpointattaches randomness toparticles:

eachparticle(x,i)(i-th particle starting atx) has a “life plan”(X_t^x,i)t≥0

(that is a continuous-time RW), and a Poisson clock with rateλ; particles move according to their life plan,

when the clock of a particle rings, if it is alone then its gets asleep, and in this case its clock stops;

when a particle is awoken, its clock resumes ticking.

,→we get a whole family of paths(Y_t^x,i)t≥0, which carries more information.

Properties:Not the above, but a control on the effect of adding one particle.

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Site fixation vs. particle fixation

Definition

Site fixationoccurs when, at each site, there is eventually no active particle.

Particle fixationoccurs when each particle is eventually sleeping.

Example of use.Assume particles fixate a.s., then µ=E[# particles initially at 0]

=E[# sites where a particle initially at 0 settles]

=

∑

v

P(some particle initially at 0 settles atv)

=

∑

v

P(some particle initially at−vsettles at 0)

=E[# particles settling at 0]≤1. Theorem (Amir–Gurel-Gurevich, 2012)

Site fixation implies particle fixation. Thus, they are equivalent. Andµc≥1. (for i.i.d. initial conditions, 0-1 laws hold for site and particle fixation)

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Site fixation vs. particle fixation

Definition

=

∑

v

=

∑

v

=E[# particles settling at 0]≤1.

Theorem (Amir–Gurel-Gurevich, 2012)

Site fixation implies particle fixation. Thus, they are equivalent. Andµc≥1. (for i.i.d. initial conditions, 0-1 laws hold for site and particle fixation)

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Site fixation vs. particle fixation

Definition

=

∑

v

=

∑

v

=E[# particles settling at 0]≤1.

Theorem (Amir–Gurel-Gurevich, 2012)

Site fixation implies particle fixation. Thus, they are equivalent. Andµc≥1.

(for i.i.d. initial conditions, 0-1 laws hold for site and particle fixation)

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A non-fixation condition

•Direct technique for proving non-fixation: finding a strategy that makes arbitrarily many particlesvisit preciselythe siteo.

•In fact, making a positive density of particlesexit a boxis sufficient.

Consider an ARW with i.i.d. initial configuration.

Forn∈N, letV_n={−n, . . . ,n}^d, denoteP[Vn]the law of the ARW restricted toV_n(i.e. particles freeze outside),M_nthe number of particles exitingV_n. Proposition

lim sup_n^E^[Vn]_|V^[Mⁿ^]

n| >0 ⇒ (particle) non-fixation, a.s.

LetVe_n=Vn−logn. Then, ifη0(x)≤Ka.s. (to simplify) E[Mn]≤µ|V_n\Ve_n|+

∑

x∈eVn,i∈N

P(i≤η0(x), particleY^x,iexitsV_n)

≤o(|Vn|) +|eV_n|KP(particleY^0,1reaches distance logn) P(Y^0,1does not fixate) =lim_nP(Y^0,1reaches dist. logn)≥lim sup_n^E[M_|Vⁿ^]

n|

Why do we haveE[Mn]≥E[Vn][Mn]?

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A non-fixation condition

∑

x∈eVn,i∈N

n|

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A non-fixation condition

∑

x∈eVn,i∈N

n|

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A monotonicity lemma

Usualsite-wise monotonicity: adding particles increases the number of used instructions (topplings).

We need avariantto distinguish particles exitingV_n. Let us color inBluethe particles that start inV_n, inRedthe particles that start outsideV_n, and modify the site-wise construction as follows:

BlueandReduseindependentstacks of instructions

(but active particles awaken any sleepy particle, hence an interaction) When aBlueparticle exitsVn, it becomesRed.

,→Colorblind dynamics still is ARW. Then monotonicity extends, in two steps:

•With onlyBlueparticles at the beginning,notfreezingRedparticles outside ofV_nanymore increases the number of usedBlueinstructions, and thusM_n: (denoting the law of this process byPVn)

E[Vn][Mn]≤EVn[Mn]

•AddingRedparticles at the beginning still increases the number of used Blueinstructions, and thusMn:

EVn[Mn]≤E[Mn]

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A monotonicity lemma

Usualsite-wise monotonicity: adding particles increases the number of used instructions (topplings).We need avariantto distinguish particles exitingV_n. Let us color inBluethe particles that start inV_n, inRedthe particles that start outsideV_n, and modify the site-wise construction as follows:

,→Colorblind dynamics still is ARW.

Then monotonicity extends, in two steps:

E[Vn][Mn]≤EVn[Mn]

EVn[Mn]≤E[Mn]

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A monotonicity lemma

Usualsite-wise monotonicity: adding particles increases the number of used instructions (topplings).We need avariantto distinguish particles exitingV_n. Let us color inBluethe particles that start inV_n, inRedthe particles that start outsideV_n, and modify the site-wise construction as follows:

,→Colorblind dynamics still is ARW.

Then monotonicity extends, in two steps:

E[Vn][Mn]≤EVn[Mn]

EVn[Mn]≤E[Mn]

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Non-fixation for biased ARW on Z

^d

Letv∈R^dand assumeµ>1−F_v(λ).

Consider ARW restricted toV_n(particles freeze outside), with site-wise construction. Let us devise atoppling strategythat throws a positive density of particles outside ofV_n.

Preliminary step: levelling

Topple sites inV_nuntil all particles are either alone or outsideV_n.

LabelV_n={x₁, . . . ,x_r}so thatx₁·v≤ · · · ≤x_r·v. Main step

Fori=1, . . . ,r, if there is a particle inx_i, then topple it, and topple it again, and so on until either it exitsV_n, falls asleep onx_i+{x:x·v≤0}or reaches an empty site in{x_i+1, . . . ,x_r}.

The probability of the middle case is lower thanFv(λ), and otherwise the number of particles outsideV_nor in{x_i+1, . . . ,x_r}increases by 1. Hence,

E[Vn][Mn]≥µ|V_n| −(1−Fv(λ))|Vn|.

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Non-fixation for biased ARW on Z

^d

Topple sites inV_nuntil all particles are either alone or outsideV_n. LabelV_n={x₁, . . . ,x_r}so thatx₁·v≤ · · · ≤x_r·v.

Main step

E[Vn][Mn]≥µ|V_n| −(1−Fv(λ))|Vn|.

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Non-fixation for biased ARW on Z

^d

Topple sites inV_nuntil all particles are either alone or outsideV_n. LabelV_n={x₁, . . . ,x_r}so thatx₁·v≤ · · · ≤x_r·v.

Main step

E[Vn][Mn]≥µ|V_n| −(1−F_v(λ))|Vn|.

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Construction of the infinite-volume particle-wise process

How to proveexistenceof the ARW with infinite initial condition?

→for the usual process(ηt(·))t≥0on{0,S,1, . . .}^Z^d, the standard theory from particle systems adapt (cf. Liggett, and Andjel on Zero-Range-Process)

→for the fully-labeled system of walks, no standard reference. Also, we need to prove the existence of the previous particle-wise construction specifically. Let us sketch a probabilistic proof of existence.

What do we need to prove?Consider the following:

•η₀a finite initial configuration,

•X= (X^(x,i);x∈Z^d,i∈N)a family of (putative) paths,

•P= (P^(x,i);x∈Z^d,i∈N)a family of PP (clocks).

Letη_t(z;η₀,X,P) =set of the labels “(x,i)” of the particles atzat timet. Choose a sequence of finite subsetWn↑Z^d.

For an infiniteη0, the particle-wise construction of the ARW from (η0,X,P)iswell-definedif, for allz∈Z^d,T>0,w∈Z^d, the sequence

η_|

[0,T](z;η₀·1_w+W_n,X,P), n∈N,

is eventually constant, and its limit does not depend onw∈Z^d.

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Construction of the infinite-volume particle-wise process

How to proveexistenceof the ARW with infinite initial condition?

→for the usual process(ηt(·))t≥0on{0,S,1, . . .}^Z^d, the standard theory from particle systems adapt (cf. Liggett, and Andjel on Zero-Range-Process)

→for the fully-labeled system of walks, no standard reference. Also, we need to prove the existence of the previous particle-wise construction specifically. Let us sketch a probabilistic proof of existence.

What do we need to prove?Consider the following:

•η₀a finite initial configuration,

•X= (X^(x,i);x∈Z^d,i∈N)a family of (putative) paths,

•P= (P^(x,i);x∈Z^d,i∈N)a family of PP (clocks).

Letη_t(z;η₀,X,P) =set of the labels “(x,i)” of the particles atzat timet.

Choose a sequence of finite subsetWn↑Z^d.

For an infiniteη0, the particle-wise construction of the ARW from (η0,X,P)iswell-definedif, for allz∈Z^d,T>0,w∈Z^d, the sequence

η_|

[0,T](z;η₀·1_w+W_n,X,P), n∈N,

is eventually constant, and its limit does not depend onw∈Z^d.

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Influence of a particle

Givenη₀,X,P, the particle(x,i)has an influence onz∈Z^dduring[0,t]if removing this particle changes the fully-labeled processη_|_[0,t]×{z}(η0,X,P).

To prove well-definedness atz, we have to ensure that, for afinitenumber of n’s, some site inW_n+1\W_nhas an influence onz. The key is the following.

Lemma

Let Z^x,i_t (η0,X,P)be the set of sites influenced by(x,i)before t.

There is a branching r.w.Z one Z^dsuch that, for any given finite config.π, Z_t^x,i(π,X,P)⊂_st.x+eZt,

and E[|eZ_t|]≤e^ct.

Theorem

Assumesup_xE[η0(x)]<∞. Then the particle-wise ARW is a.s. well-defined.

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Conclusion

Extensions of parts of the proof, of possible independent interest:

The non-fixation condition naturally extends to amenable graphs (assuming|∂V_n|=o(|Vn|), positive density of exits⇒non-fixation).

The particle-wise construction extends to transitive graphs with a unimodular subgroup of automorphisms that preserves the jump distribution (needsmass transport principle).

Most striking open questions:

in the symmetric case, non-fixation for someµ<1? (even whend=1) in the biased case, fixation for someµ>0?(for symmetric case, see Sidoravicius-Teixeira 2014)