CEREMADE, CNRS, Université Paris-Dauphine, Université PSL（巴黎第九大学数学与决策研究中心）；University of Washington（华盛顿大学）；College of Staten Island (CUNY)（纽约市立学院）；CUNY Graduate Center（纽约市立大学研究生中心）；The Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Universidade de São Paulo（圣保罗大学）
Nicolas Forien, Christopher Hoffman, Tobias Johnson, Josh Meisel, Jacob Richey, Leonardo T. Rolla
We show that Activated Random Walk on $\mathbb{Z}$ is explosive above criticality. That is, activating a single particle in a supercritical state of sleeping particles triggers an infinite avalanche of activity with positive probability. This extends the same result recently proven by Brown, Hoffman, and Son for i.i.d. initial distributions to the setting of ergodic ones, thus completing the proof of a conjecture of Rolla's in dimension one. As a corollary we obtain that, for supercritical ergodic initial distributions with any positive density of particles initially active, the system will stay active almost surely. Our result is another piece of evidence attesting to the universality of the phase transition of Activated Random Walk on $\mathbb{Z}$.
Unlike many particle systems, Activated Random Walk has nontrivial behavior even in one spatial dimension. We prove inner and outer bounds on the spread of n activated random walkers from a single source in Z. The inner bound involves a comparison with the stationary distribution of activated random walkers on a finite interval, while the outer bound involves a comparison with the stabilization of an infinite Bernoulli configuration of activated random walkers on Z.
We show that for small enough sleep rate, the critical density of the symmetric Activated Random Walk model on $\mathbb Z^2$ is strictly less than one.
We prove that the model of Activated Random Walks on Z^d with biased jump distribution does not fixate for any positive density, if the sleep rate is small enough, as well as for any finite sleep rate, if the density is close enough to 1. The proof uses a new criterion for non-fixation. We provide a pathwise construction of the process, of independent interest, used in the proof of this non-fixation criterion.
We consider symmetric activated random walks on $\mathbb{Z}$, and show that the critical density $ζ_c$ satisfies $c\sqrtλ \leq ζ_c(λ) \leq C \sqrtλ$ where $λ$ denotes the sleep rate.
It has been conjectured that the critical density of the Activated Random Walk model is strictly less than one for any value of the sleeping rate. We prove this conjecture on $\mathbb{Z}^d$ when $d \geq 3$ and, more generally, on graphs where the random walk is transient. Moreover, we establish the occurrence of a phase transition on non-amenable graphs, extending previous results which require that the graph is amenable or a regular tree.
We consider Activated Random Walks on $\Z$ with totally asymmetric jumps and critical particle density, with different time scales for the progressive release of particles and the dissipation dynamics. We show that the cumulative flow of particles through the origin rescales to a pure-jump self-similar process which we describe explicitly.
In this article, we study the existence of an absorbing-state phase transition of an Abelian process that generalises the Activated Random Walk (ARW). Given a vertex transitive $G=(V,E)$, we associate to each site $x \in V$ a capacity $w_x \ge 0$, which describes how many inactive particles $x$ can hold, where $\{w_x\}_{x \in V}$ is a collection of i.i.d random variables. When $G$ is an amenable graph, we prove that if $\mathbb E[w_x]<\infty$, the model goes through an absorbing state phase transition and if $\mathbb E[w_x]=\infty$, the model fixates for all $λ>0$. Moreover, in the former case…
We consider a 1-dimensional Brownian motion whose diffusion coefficient varies when it crosses the origin. We study the long time behavior and we establish different regimes, depending on the variations of the diffusion coefficient: emergence of a non-Gaussian mul-tipeaked probability distribution and a dynamical transition to an absorbing static state. We compute the generator and we study the partial differential equation which involves its adjoint. We discuss global existence and blow-up of the solution to this latter equation.
In this paper we present rigorous results on the critical behavior of the Activated Random Walk model. We conjecture that on a general class of graphs, including $\mathbb{Z}^d$, and under general initial conditions, the system at the critical point does not reach an absorbing state. We prove this for the case where the sleep rate $λ$ is infinite. Moreover, for the one-dimensional asymmetric system, we identify the scaling limit of the flow through the origin at criticality. The case $λ< + \infty$ remains largely open, with the exception of the one-dimensional totally-asymmetric case, for which…
We study the dynamics of two conservative lattice gas models on the infinite d-dimensional hypercubic lattice: the Activated Random Walks (ARW) and the Stochastic Sandpiles Model (SSM), introduced in the physics literature in the early nineties. Theoretical arguments and numerical analysis predicted that the ARW and SSM undergo a phase transition between an absorbing phase and an active phase as the initial density crosses a critical threshold. However a rigorous proof of the existence of an absorbing phase was known only for one-dimensional systems. In the present work we establish the existe…
We consider Activated Random Walks on arbitrary finite networks, with particles being inserted at random and absorbed at the boundary. Despite the non-reversibility of the dynamics and the lack of knowledge on the stationary distribution, we explicitly determine the relaxation time of the process, and prove that separation cutoff is equivalent to the product condition. We also provide sharp estimates on the center and width of the cutoff window. Finally, we illustrate those results by establishing explicit separation cutoffs on various networks, including: (i) large finite subgraphs of any fix…
We prove that for the Activated Random Walks model on transitive unimodular graphs, if there is fixation, then every particle eventually fixates, almost surely. We deduce that the critical density is at most 1. Our methods apply for much more general processes on unimodular graphs. Roughly put, our result apply whenever the path of each particle has an automorphism invariant distribution and is independent of other particles' paths, and the interaction between particles is automorphism invariant and local. This allows us to answer a question of Rolla and Sidoravicius, in a more general setting…
Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framew…
We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density $μ_c$ for sustained activity is strictly between 0 and 1. It was known that $μ_c>0$ on $\mathbb{Z}^d$, $d\geq 1$, and that $μ_c<1$ on $\mathbb{Z}$ for small enough sleeping rate. We show that $μ_c\to 0$ as $λ\to 0$ in all vertex-transitive transient graphs, implying that $μ_c<1$ for small enough sleeping rate. We also show that $μ_c<1$ for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we…
We consider Activated Random Walk (ARW), a particle system with mass conservation, on the cycle $\mathbb{Z}/n\mathbb{Z}$. One starts with a mass density $μ>0$ of initially active particles, each of which performs a simple symmetric random walk at rate one and falls asleep at rate $λ>0.$ Sleepy particles become active on coming in contact with other active particles. There have been several recent results concerning fixation/non-fixation of the ARW dynamics on infinite systems depending on the parameters $μ$ and $λ$. On the finite graph $\mathbb{Z}/n\mathbb{Z}$, unless there are more than $n$ p…
We consider the activated random walk (ARW) model where particles follow the path of a general Markov process on a general graph. We prove ARW dominates a simpler process, multiple source internal aggregation (MSIA), and use this to formulate a deterministic sufficient condition on initial occupations for nonfixation of ARW and similar variants. In particular, on bounded degree graphs, initial occupation density greater than one almost surely implies nonfixation, where independence requirements are weakened to ergodic in the case of Euclidean lattices. We show that for Euclidean lattices of di…
We consider the activated random walk (ARW) model on $\mathbb{Z}^d$, which undergoes a transition from an absorbing regime to a regime of sustained activity. In any dimension we prove that the system is in the active regime when the particle density is less than one, provided that the jump distribution is biased and that the sleeping rate is small enough. This answers a question from Rolla and Sidoravicius (2012) and Dickman, Rolla and Sidoravicius (2010) in the case of biased jump distribution. Furthermore, we prove that the critical density depends on the jump distribution.
* ACTIVATED RANDOM WALK MODEL * This is a conservative particle system on the lattice, with a Markovian continuous-time evolution. Active particles perform random walks without interaction, and they may as well change their state to passive, then stopping to jump. When particles of both types occupy the same site, they all become active. This model exhibits phase transition in the sense that for low initial densities the system locally fixates and for high densities it keeps active. Though extensively studied in the physics literature, the matter of giving a mathematical proof of such phase tr…
We consider a non-homogeneous random walks system on $\bbZ$ in which each active particle performs a nearest neighbor random walk and activates all inactive particles it encounters up to a total amount of $L$ jumps. We present necessary and sufficient conditions for the process to survive, which means that an infinite number of random walks become activated.
The frog model is an interacting particle system on a graph. Active particles perform independent simple random walks, while sleeping particles remain inert until visited by an active particle. Some number of sleeping particles are placed at each site sampled independently from a certain distribution, and then one particle is activated to begin the process. We show that the recurrence or transience of the model is sensitive not just to the expectation but to the entire distribution. This is in contrast to closely related models like branching random walk and activated random walk.
We study a particle system with hopping (random walk) dynamics on the integer lattice $\mathbb Z^d$. The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate $λ> 0$, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density $ζ$ and the …
In the stochastic sandpile model on a graph, particles interact pairwise as follows: if two particles occupy the same vertex, they must each take an independent random walk step with some probability $0<p<1$ of not moving. These interactions continue until each site has no more than one particle on it. We provide a formal coupling between the stochastic sandpile and the activated random walk models, and we use the coupling to show that for the stochastic sandpile with $n$ particles on the cycle graph $\mathbb{Z}_n,$ the system stabilizes in $O(n^3)$ time for all initial particle configurations…
We study an interacting random walk system on Z where at time 0 there is an active particle at 0 and one inactive particle on each site $n \ge 1$. Particles become active when hit by another active particle. Once activated, the particle starting at n performs an asymmetric, translation invariant, nearest neighbor random walk with left jump probability $l_n$. We give conditions for global survival, local survival and infinite activation both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan (with parameter depending on the start…
We propose an approach to construct Bernoulli trials $\{X_i, i\ge 1\}$ combining dependence and independence periods, and call it Bernoulli sequence with random dependence (BSRD). The structure of dependence, on the past $S_i = X_1 + \ldots + X_i$, {defines} a class of non-Markovian random walks of recent interest in the literature. In this paper, the dependence is activated by an auxiliary collection of Bernoulli trials $\{Y_i, i\ge 1\}$, called {\it memory switch sequence}. We introduce the concept of {\it memory lapses property}, which {is} characterized by intervals of consecutive independ…
We study the long-time behavior of conservative interacting particle systems in $Z$: the activated random walk model for reaction-diffusion systems and the stochastic sandpile. We prove that both systems undergo an absorbing-state phase transition.
Consider a growing system of random walks on the 3,2-alternating tree, where generations of nodes alternate between having two and three children. Any time a particle lands on a node which has not been visited previously, a new particle is activated at that node, and begins its own random walk. The model described belongs to a class of problems that are collectively referred to as the frog model. Building on a recent proof of recurrence (meaning infinitely many frogs hit the root with probability one) on the regular binary tree, this paper establishes recurrence for the 3,2-alternating case.
We examine a system of interacting random walks with leftward drift on $\mathbb{Z}$, which begins with a single active particle at the origin and some distribution of inactive particles on the positive integers. Inactive particles become activated when landed on by other particles, and all particles beginning at the same point posses equal leftward drift. Once activated, the trajectories of distinct particles are independent. This system belongs to a broader class of problems involving interacting random walks on rooted graphs, referred to collectively as the frog model. Additional conditions …
We consider the Activated Random Walk model in any dimension with any sleep rate and jump distribution and ergodic initial state. We show that the stabilization properties depend only on the average density of particles, regardless of how they are initially located on the lattice.
A two-type version of the frog model on $\mathbb{Z}^d$ is formulated, where active type $i$ particles move according to lazy random walks with probability $p_i$ of jumping in each time step ($i=1,2$). Each site is independently assigned a random number of particles. At time 0, the particles at the origin are activated and assigned type 1 and the particles at one other site are activated and assigned type 2, while all other particles are sleeping. When an active type $i$ particle moves to a new site, any sleeping particles there are activated and assigned type $i$, with an arbitrary tie-breaker…
We consider Activated Random Walk (ARW), a model which generalizes the Stochastic Sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on $\mathbb{Z}$ with mass conservation. One starts with a mass density $μ>0$ of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate $λ>0$. Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process, and show for small enough $λ$, the critical mass density for f…
The frog model is a system of interacting random walks. Initially, there is one particle at each vertex of a connected graph $\mathcal{G}$. All particles are inactive at time zero, except for the one which is placed at the root of $\mathcal{G}$, which is active. At each instant of time, each active particle may die with probability $1-p$. Once an active particle survives, it jumps on one of its nearest vertices, chosen with uniform probability, performing a discrete time simple symmetric random walk (SRW) on $\mathcal{G}$. Up to the time it dies, it activates all inactive particles it hits alo…
The frog model is a system of random walks where active particles set sleeping particles in motion. On the complete graph with n vertices it is equivalent to a well-understood rumor spreading model. We given an alternate and elementary proof that the wake-up time, i.e. the expected time for every particle to be activated, is Theta(log n). Additionally, we give an explicit distributional equation for the wakeup time as a weighted sum of geometric random variables. This project was part of the University of Washington Research Experience for Undergraduates program.
