The Critical Density for Activated Random Walks is always less than 1

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arXiv:2210.04779v2 [math.PR] 12 Jul 2024

Submitted to the Annals of Probability

THE CRITICAL DENSITY FOR ACTIVATED RANDOM WALKS IS ALWAYS

LESS THAN 1

BYAMINE ASSELAH1,a, NICOLAS FORIEN2,bAND ALEXANDRE GAUDILLIÈRE3,c

1LAMA, UPEC, UPEM, CNRS, F-94010, Créteil, France and NYU Shanghai, aamine.asselah@u-pec.fr

2Sapienza Università di Roma, Dipartimento di Matematica, Roma, Italy, bnicolas.forien@uniroma1.it

3Aix Marseille Univ, CNRS, I2M, Marseille, France , calexandre.gaudilliere@math.cnrs.fr

Activated Random Walks, on Zdfor any d>1, is an interacting particle

system, where particles can be in either of two states: active or frozen. Each

active particle performs a continuous-time simple random walk during an

exponential time of parameter λ, after which it stays still in the frozen state,

until another active particle shares its location, and turns it instantaneously

back into activity. This model is known to have a phase transition, and we

show that the critical density, controlling the phase transition, is less than one

in any dimension and for any value of the sleep rate λ. We provide upper

bounds for the critical density in both the small λand large λregimes.

Keywords and phrases. Activated random walks, phase transition, self-

organized criticality.

MSC 2020 subject classiﬁcations. 60K35, 82B26.

1. Model and results.

1.1. Activated Random Walks. This paper is a companion to [9]. We continue our study

of a speciﬁc reaction-diffusion model known as Activated Random Walks (ARW) invented

to study self-organized criticality. Informally, random walks diffuse on a graph which has

a tendency to hinder the motion of lonely walkers, whereas the vicinity of other diffusions

turns hindered particles into diffusive ones. Here, we consider the Euclidean lattice Zdin any

dimension, or a large Euclidean torus. The initial conﬁguration is an independent Poisson

number of particles at each site of Zd, with parameter µ < 1. Each particle can be in any

of two states: active or frozen (or sleeping). Each active particle performs a continuous-time

simple random walk with rate 1and is equipped with an independent exponential clock of

parameter λ, at the marks of which the particle changes state, and stops moving. When a

frozen particle shares a site with another particle, it gets instantly activated. When the graph

is an inﬁnite Euclidean lattice, and one increases the initial density of active particles, one

expects to see a transition, at a critical density µc(λ), from a regime of low density where

particles are still to a high density regime of conﬁgurations made of constantly evolving

islands of sleeping particles at low density in a sea of diffusing particles at high density.

When we start with a large number of active particles at the origin we expect a large ball to be

eventually covered at density µc(λ). This phenomenon is known as self-organized criticality,

in the sense that the system alone reaches a critical state. This notion was introduced in

the eighties by Bak, Tang and Wiesenfeld [3] together with a related toy model, the abelian

sandpile. The ARW model, which is less constrained, was actually popularized some 13 years

ago by our late friend Vladas Sidoravicius and we refer to Levine and Liang [15] for some

comparison between the two models. The ARW model shares with the abelian sandpile the

MSC2020 subject classiﬁcations:Primary 60K35, 82B26.

Keywords and phrases: Activated random walks, Self-organized criticality.

nice feature that the order in which the particles are launched is irrelevant, which is known

as the abelian property.

When working with Zd, and when active particles are drawn from a product Poisson mea-

sure of intensity µat each site, ARW is known to have a phase transition between an active

phase and a frozen phase. The active phase is characterized by every vertex being visited in-

ﬁnitely many times, whereas in the frozen phase the origin is visited a ﬁnite number of times.

In a seminal work Rolla and Sidoravicius [19] prove that the system stays active forever with

a probability which is increasing in µ, and which satisﬁes a zero-one law under Pλ

µ, law of

the process when the sleep rate is λand the initial conﬁguration is drawn from the product

Poisson measure of intensity µ. Thus, the following density µc(λ)is well deﬁned:

µc(λ) = inf nµ:Pλ

µ(the origin is visited a ﬁnite number of times) = 0 o.

In [20] Rolla, Sidoravicius and Zindy show that µc(λ)is the same number when the initial

conﬁguration is drawn from any translation-invariant ergodic measure with mean µ. On Zd

with d>3, Stauffer and Taggi in [21] show that when λis small, µc(λ)<1, and they pro-

vide a general lower bound µc(λ)>λ/(1 + λ), which is valid in any dimension and for

every λ > 0. In a subsequent work [22] Taggi shows that µc(λ)<1for all λ∈(0,∞)on Zd

with d>3, and provides an upper bound on the critical density, showing that µc(λ)6Cd√λ

for every λ > 0, for some positive constant Cd.

Even in dimension one, ARW is far from trivial. In d= 1, Basu, Ganguly and Hoffman

introduce in [4] a block dynamics allowing them to replace the complex correlation of the

odometer function (measuring the number of instructions used at each site) by some balance

equations at the end-points of their blocks (where particles leave) and at the centers (where

particles arrive). They obtain that µc(λ)<1for small λ. Following the same approach, As-

selah, Rolla and Schapira show in [2] that µc(λ) = O(√λ)for small λ, and Hoffman, Richey

and Rolla show in [10] that, for any λ,µc(λ)<1in dimension 1. Finally, let us mention that

a lot of works have considered asymmetric random walks, where µc(λ)<1has been settled.

In dimension 2, two independent recent works by Forien and Gaudillière [9] on the one hand,

and Y. Hu [11] on the other hand, have established that µc(λ)<1when λis small enough.

The family of ARW models is very rich as we vary λfrom 0to ∞, and as we vary the

initial conditions with active and frozen particles. Let us illustrate this with two examples.

When λ= 0, the active particles stay alive forever: if we start with a product Poisson dis-

tribution of sleeping particles and one active particle at the origin, the model is known as

the frog model. Kesten and Sidoravicius have also studied a model for the propagation of

an infection, where the sleeping particles can move at a slower rate than the active ones.

When λ=∞, and we send active particles from the origin, then this is the celebrated model

called internal diffusion limited aggregation (IDLA). This latter model is much older, and in

a sense much simpler since frozen particles remain so forever. It has been thoroughly stud-

ied, and a shape theorem has been obtained in the nineties by [13] on Zd, and on a few other

interesting graphs, as well as for closely related variants: uniform IDLA [5], Hasting-Levitov

dynamics [17], rotor-router [16] and divisible sandpiles. Besides, ﬂuctuations around the typ-

ical shape have been obtained on Zdindependently by Asselah and Gaudillière in [1] and by

Jerison, Levine and Shefﬁeld in [12]. It remains the focus of recent interest [6].

We show in this paper that when we start with particles which are all active on the

torus Zd

n:= (Z/nZ)d, for any dimension dand λ > 0, there is a density µ < 1, independent

of n, above which the system remains active during an exponentially large time (in |Zd

n|) with

overwhelming probability. This implies that there exists a non-trivial active phase on the in-

ﬁnite Euclidean lattice for all sleep rates λ. In other words, when we start the system on Zd

with active particles distributed as a product measure with more than µc(λ)<1particles per

site on average, then the origin is almost surely visited inﬁnitely many times.

CRITICAL DENSITY FOR ARW 3

1.2. Main results. Our main result is the following.

THEOREM 1.1. In any dimension d>1, for every sleep rate λ > 0, the critical density

of the Activated Random Walks model on Zdsatisﬁes µc(λ)<1.

This result is new in d= 2 and our proof encompasses all dimensions (with small changes).

Since it is constructive, it does provide upper bounds on µc(λ): these bounds are new in d= 2,

and improve existing bounds for d>3. Note that we cover the regime of large λas well as

small λ. Theorem 1.1 follows from the upper bounds we now present (and from the fact that

the critical density µcis non-decreasing in λ, see for example [18]).

THEOREM 1.2. In dimension d= 2, there exists a > 0such that, for λsmall enough,

(1.1) µc(λ)6λ|ln λ|a,

and there exists c > 0such that, for λlarge enough,

(1.2) µc(λ)61−c

λ(ln λ)2.

THEOREM 1.3. In dimension d>3, there exists c=c(d)>0such that, for λsmall

enough,

(1.3) µc(λ)6c λ ,

and there exists c=c(d)>0such that, for λlarge enough,

(1.4) µc(λ)61−c

λln λ.

These bounds, in the two regimes of sleep rate, are of a correct nature (up to some loga-

rithmic factor in λ), as we try to justify in our heuristic discussion below.

REMARK. Our proof method also works to show that µc(λ)<1for every λ > 0in di-

mension 1, with minor changes, but in this setting it does not yield signiﬁcantly new bounds

on the critical density, hence we choose not to detail this case. We limit ourselves to some

comments in section 1.6 about how to adapt our proof to the one-dimensional case.

1.3. Heuristics. At a heuristic level, the critical density of frozen particles can be thought

of as follows. Consider a conﬁguration of frozen particles drawn from a Poisson product

measure on Zdwith density µ, and launch an active particle at the origin. The density µ

equals µc(λ)if an active particle, in its journey before freezing, encounters on average exactly

one sleeping particle. In other words, the number of active particles should strike a balance:

one particle wakes up when another one freezes. Thus, if Rtis the number of distinct visited

sites in a time period [0, t]by a continuous-time random walk, and if τis an independent

exponential time of mean 1/λ, we expect that

(1.5) E[Rτ]·µc(λ)≃1.

The symbol ≃is here to remind the reader that this is just heuristics. Now, there are two

regimes: small λwhere τis of order of its mean 1/λ which is large, and the regime of

large λwhere the particle makes a jump with probability 1/(1 + λ)which is small. In the

case of small λ, we rewrite (1.5) as E[R1/λ]·µc(λ)≃1, and we only need to recall the large

time asymptotics of the range of a random walk (see for example [8]):

E[Rt] = 





O(√t)if d= 1

O(t

ln t)if d= 2

O(t)if d>3.

This implies the following heuristics for µc(λ)for small λ:

µc(λ) = 





O(√λ)if d= 1

O(λ|ln λ|)if d= 2

O(λ)if d>3.

When λis large, the active particle makes one jump with probability 1/(1 + λ)so that

E[Rτ]≃1 + 1

λ,

which, plugged into (1.5), suggests that µc(λ)≃1−1/λ when λ→∞.

Establishing these bounds remains a challenging problem, as well as establishing some

shape theorem, or understanding ARW at the critical density. The model of ARW presents

many other interesting questions, and we refer to Rolla’s survey [18] for a nice review. One

difﬁculty is that the time a particle stays in one of its two states actually depends on the

local density of particles which itself changes with time: if an active particle travels amidst

a region of high density, then it most likely remains active as long as it remains inside this

region; instead, if it crosses a low density region, it most likely switches to a frozen state at

the ﬁrst mark of its exponential clock.

1.4. Sketch of the proof of Theorem 1.2.Most of our work is devoted to proving The-

orem 1.2 (the case d= 2), which is our main result, while Theorem 1.3 (the transient

case d>3) requires much less technology and simply follows as a by-product of an in-

termediate Lemma.

We now describe informally the six steps of our strategy, of which three are taken from [9],

and three are new. The new ideas, namely the dormitories, the ping-pong rally and the

coloured loops, are all of a hierarchical nature.

To show that it takes an exponentially large time to stabilize a conﬁguration on the torus,

we introduce a hierarchical structure on the set where the particles eventually settle, which

we call the hierarchical dormitory. With this construction, we ﬁrst show that some elemen-

tary blocks of this hierarchy, called the clusters, have a stabilization time exponentially large

in their size. We then perform an induction using a toppling strategy which we call the ping-

pong rally, where neighbouring clusters interact and reactivate each other many times, lead-

ing to a stabilization time for their union which is roughly the product of the individual

stabilization times. Thus, we obtain that at each space scale, the stabilization time of a clus-

ter is of order an exponential in the volume of the cluster. The coloured loops are the last

important ingredient in our proof: modifying slightly the dynamics of the ARW model by

ignoring some reactivation events, we are able to obtain some independence between the

different levels of the hierarchy, which turns out to be crucial in our inductive proof.

1.4.1. Working on the torus and using the abelian property. In [9] it is shown that µc<1

if, when starting with a density µ < 1of active particles, the time needed to stabilize ARW on

the torus Zd

n= (Z/nZ)dis exponentially large in nwith high probability. Thus, we always

consider Zd

nwith nlarge enough.

CRITICAL DENSITY FOR ARW 5

We recall that there is a celebrated graphical representation, known as the site-wise or

Diaconis-Fulton representation (see [7] for the original construction or [18] for a nice pre-

sentation in the context of ARW), where we pile stacks of independent instructions on top of

each site of Zd

n, and use these instructions one after another to move the particles.

A key property of this representation of the model, known as the abelian property, is that

the ﬁnal conﬁguration and the number of steps performed (these steps are called topplings)

do not depend on the order with which the instructions are used, allowing us to choose an

arbitrary strategy to move particles. We rely heavily on this property throughout our work

(although our graphical representation, described below in section 2.5 is no longer abelian,

its construction itself relies on the abelian property of the ARW model).

1.4.2. Decomposition over all possible settling sets. Whereas [9] focuses on the case

where the sleep rate λis small, in our case we ﬁx λ, which can be either small or large, and

we look for a density µ < 1so that the stabilization time on the torus is exponentially large.

Starting from a ﬁxed initial conﬁguration with µndparticles, we decompose the probability

to stabilize the conﬁguration in a given time, summing over all the possible sets where the

particles can settle.

Then, as in [9], for a ﬁxed couple (λ, µ)we look for an estimate on the stabilization time

which is uniform over all possible settling sets A⊂Zd

n, and we perform a union bound.

Hence, in our estimate we get a combinatorial factor nd

µndcorresponding to the number of

possible settling sets. This factor can be thought of as an “entropic” term, which we have

to outweigh by an “energy” term corresponding to the probability to stabilize in a given

set A⊂Zd

nin a short time. Since this entropic factor nd

µndgets smaller when µis either close

to 0or close to 1, we concentrate on these two distinct regimes of the sleep rate, namely λ→0

and λ→ ∞, with a corresponding density µ→0or µ→1.

The use of a uniform estimate over settling sets is responsible for some logarithm factors

appearing in the bounds that we obtain on the critical density µc. This can be seen in the state-

ment of Lemma 2.1 (where ψ(µ)corresponds to the entropy, while κcontrols the energy) and

in the ﬁnal estimates for our proofs in sections 3and 4, where we tune the parameters λand µ

such that the entropy-energy balance is favourable. Thus, a possible direction to improve our

estimates on µccould be to reﬁne this union bound by ruling out some sets Aon which it is

very unlikely that the particles settle.

1.4.3. Reduction to a model with density 1 on the trace graph. Once such a set A⊂Zd

nis

ﬁxed, we look for an upper bound on the probability that the particles settle on Ain a given

time. To this end, as explained in section 3.1 of [9], we suppress all sleeping instructions

on Zd

n\A. Indeed, provided that particles eventually settle in A, these sleeping instructions

are overridden at some time or another. Then, using the abelian property of the model, we

may ﬁrst let each particle move until it reaches an empty site of A. Thus, we may start from

the conﬁguration with exactly one active particle on each site of A(see section 3.2 of [9]).

Thus, we end up with a simpliﬁed model on a ﬁxed subset Athat we call the dormitory, which

starts fully occupied with active particles. These active particles cannot settle anywhere but

on A.

Let us now describe the order with which the particles move. At the beginning of each step,

we choose an active particle and read its ﬁrst unused instruction. If it is a sleep instruction,

the step is over, the particle falls asleep and we choose another particle at the next step.

Otherwise, if it is a jump instruction, we let the particle jump, and follow instructions along

its path until it goes back to its starting point. Indeed, when the particle is not at its starting

position, it is either outside of A, and there is no sleep instruction, or it is on top of another

particle, and the sleep instructions have no effect. Note that at the end of its loop, the particle

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arXiv:2210.04779v2[math.PR]12Jul2024SubmittedtotheAnnalsofProbabilityTHECRITICALDENSITYFORACTIVATEDRANDOMWALKSISALWAYSLESSTHAN1BYAMINEASSELAH1,a,NICOLASFORIEN2,bANDALEXANDREGAUDILLIÈRE3,c1LAMA,UPEC,UPEM,CNRS,F-94010,Créteil,FranceandNYUShanghai,aamine.asselah@u-pec.fr2SapienzaUniversitàdiRoma,Dipart...

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