Independent Simple Random Walks Research Articles

A multivariate extension of the Erdős–Taylor theorem

The Erdős–Taylor theorem (Acta Math Acad Sci Hungar, 1960) states that if LN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{L}_N$$\\end{document} is the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then πlogNLN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{\\pi }{\\log N} \\,\ extsf{L}_N$$\\end{document} converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if LN(1,2)=∑n=1N1{Sn(1)=Sn(2)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{L}_N^{(1,2)}=\\sum _{n=1}^N \\mathbb {1}_{\\{S_n^{(1)}= S_n^{(2)}\\}}$$\\end{document}, then πlogNLN(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{\\pi }{\\log N}\\, \ extsf{L}^{(1,2)}_N$$\\end{document} converges in distribution to an exponential random variable of parameter one. We prove that for every h⩾3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h \\geqslant 3$$\\end{document}, the family {πlogNLN(i,j)}1⩽i<j⩽h\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\big \\{ \\frac{\\pi }{\\log N} \\,\ extsf{L}_N^{(i,j)} \\big \\}_{1\\leqslant i<j\\leqslant h}$$\\end{document}, of logarithmically rescaled, two-body collision local times between h independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdős–Taylor theorem. We also discuss connections to directed polymers in random environments.

Infinite collision property for the three-dimensional uniform spanning tree

Let [Formula: see text] be the three-dimensional uniform spanning tree, whose probability law is denoted by [Formula: see text]. For [Formula: see text]-a.s. realization of [Formula: see text], the recurrence of the simple random walk on [Formula: see text] is proved in Benjamini et al. (2001) [I. Benjamini, R. Lyons, Y. Peres and O. Schramm, Uniform spanning forests, Ann. Probab. 29(1) (2001) 1–65] and it is also demonstrated in Hutchcroft and Peres (2015) [T. Hutchcroft and Y. Peres, Collisions of random walks in reversible random graphs, Electron. Commun. Probab. 20(63) (2015) 1–6] that two independent simple random walks on [Formula: see text] collide infinitely often. In this paper, we will give a quantitative estimate on the number of collisions of two independent simple random walks on [Formula: see text], which provides another proof of the infinite collision property of [Formula: see text].

Percolation of worms

We introduce a new correlated percolation model on the d-dimensional lattice Zd called the random length worms model. Assume given a probability distribution on the set of positive integers (the length distribution) and v∈(0,∞) (the intensity parameter). From each site of Zd we start POI(v) independent simple random walks with this length distribution. We investigate the connectivity properties of the set Sv of sites visited by this cloud of random walks. It is easy to show that if the second moment of the length distribution is finite then Sv undergoes a percolation phase transition as v varies. Our main contribution is a sufficient condition on the length distribution which guarantees that Sv percolates for all v>0 if d≥5. E.g., if the probability mass function of the length distribution is m(ℓ)=c⋅ln(ln(ℓ))ɛ/(ℓ3ln(ℓ))1[ℓ≥ℓ0]for some ℓ0>ee and ɛ>0 then Sv percolates for all v>0. Note that the second moment of this length distribution is only “barely” infinite. In order to put our result in the context of earlier results about similar models (e.g., finitary random interlacements, loop percolation, Bernoulli hyper-edge percolation, Poisson Boolean model, ellipses percolation, etc.), we define a natural family of percolation models called the Poisson zoo and argue that the percolative behaviour of the random length worms model is quite close to being “extremal” in this family of models.

Random Walk on the Simple Symmetric Exclusion Process

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density rho in [0, 1] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities rho except for at most two values rho _-, rho _+ in [0, 1]. The asymptotic speed we obtain in our LLN is a monotone function of rho . Also, rho _- and rho _+ are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is 1/2 and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. We also prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

Continuity for the asymptotic shape in the frog model with random initial configurations

We consider the so-called frog model with random initial configurations, which is described by the following evolution mechanism of simple random walks on the multidimensional cubic lattice: Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and they independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start doing independent simple random walks. An interest of this model is how initial configurations affect the asymptotic shape of the set of all sites visited by active particles up to a certain time. Thus, in this paper, we prove continuity for the asymptotic shape in the law of the initial configuration.

Universality of local statistics for noncolliding random walks

We consider the $N$-particle noncolliding Bernoulli random walk—a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with steps $\in\{0,1\}$ by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times $T\ll N$ as $N\to+\infty$. We show that if the particle density of the initial configuration is bounded away from $0$ and $1$ down to scales $\mathsf{D}\ll T$ in a neighborhood of size $\mathsf{Q}\gg T$ of some location $x$ (i.e., $x$ is in the “bulk”), and the initial configuration is balanced in a certain sense, then the space-time local statistics at $x$ are asymptotically governed by the extended discrete sine process (which can be identified with a translation invariant ergodic Gibbs measure on lozenge tilings of the plane). We also establish similar results for certain types of random initial data. Our proofs are based on a detailed analysis of the determinantal correlation kernel for the noncolliding Bernoulli random walk. The noncolliding Bernoulli random walk is a discrete analogue of the $\boldsymbol{\beta}=2$ Dyson Brownian motion whose local statistics are universality governed by the continuous sine process. Our results parallel the ones in the continuous case. In addition, we naturally include situations with inhomogeneous local particle density on scale $T$, which nontrivially affects parameters of the limiting extended sine process, and in a particular case leads to a new behavior.

Exceptional graphs for the random walk

Une marche aleatoire simple $\mathcal{W}$ sur $\mathbb{Z}^{2}$ induit pour chaque sous-graphe couvrant $G\subset \mathbb{Z}^{2}$ une marche $\mathcal{W}_{G}$ : notamment, si on regarde $\mathcal{W}$ comme un mot infini uniforme sur l’alphabet $\{\mathbf{x},-\mathbf{x},\mathbf{y},-\mathbf{y}\}$, alors la marche $\mathcal{W}_{G}$ commence a l’origine et suit les directions definis par $\mathcal{W}$, en acceptant seulement les pas de $\mathcal{W}$ le long desquels $\mathcal{W}_{G}$ reste sur $G$. Pour chaque $G\subset \mathbb{Z}^{2}$ fixe, la marche $\mathcal{W}_{G}$ a la loi d’une marche aleatoire simple sur $G$ et alors $\mathcal{W}_{G}$ est presque surement recurrent dans le sens que $\mathcal{W}_{G}$ visite chaque sommet de $G$ connexe a l’origine un nombre infini de fois. Alors, une question naturelle surgit : est-ce que presque surement pour $\mathcal{W}$, les marches $\mathcal{W}_{G}$ sont recurrents pour tous $G\subset \mathbb{Z}^{2}$ a la fois? Dans cet article, on repond a cette question d’une maniere negative, en montrent l’existence des graphes exceptionnels sur lesquels la marche est transitoire. En fait, on montre que meme si on considere un nombre denombrable des marches aleatoires simples independantes, alors des graphes exceptionnels existent. D’autre cote, on montre que il n’existe pas des graphes exceptionnels pour la marche aleatoire branchante.

Multi-scale Lipschitz percolation of increasing events for Poisson random walks

Consider the graph induced by $\mathbb{Z}^d$, equipped with uniformly elliptic random conductances. At time $0$, place a Poisson point process of particles on $\mathbb{Z}^d$ and let them perform independent simple random walks. Tessellate the graph into cubes indexed by $i\in\mathbb{Z}^d$ and tessellate time into intervals indexed by $\tau$. Given a local event $E(i,\tau)$ that depends only on the particles inside the space time region given by the cube $i$ and the time interval $\tau$, we prove the existence of a Lipschitz connected surface of cells $(i,\tau)$ that separates the origin from infinity on which $E(i,\tau)$ holds. This gives a directly applicable and robust framework for proving results in this setting that need a multi-scale argument. For example, this allows us to prove that an infection spreads with positive speed among the particles.

Sensitivity of the frog model to initial conditions

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.

Random walks in random conductances: Decoupling and spread of infection

Let (G,μ) be a uniformly elliptic random conductance graph on Zd with a Poisson point process of particles at time t=0 that perform independent simple random walks. We show that inside a cube QK of side length K, if all subcubes of side length ℓ<K inside QK have sufficiently many particles, the particles return to stationarity after cℓ2 time with a probability close to 1. We show that in this setup, an infection spreads with positive speed in any direction. Our framework is robust enough to allow us to also extend the result to infection with recovery.

Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

The parabolic Anderson model is defined as the partial differential equation \partial u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t), x\in\Z^d, t\geq 0, where \kappa \in [0,\infty) is the diffusion constant, \Delta is the discrete Laplacian, and \xi is a dynamic random environment that drives the equation. The initial condition u(x,0)=u_0(x), x\in\Z^d, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d\kappa, split into two at rate \xi \vee 0, and die at rate (-\xi) \vee 0. Our focus is on the Lyapunov exponents \lambda_p(\kappa) = \lim_{t\to\infty} \frac{1}{t} \log \E([u(0,t)]^p)^{1/p}, p \in \N, and \lambda_0(\kappa) = \lim_{t\to\infty} \frac{1}{t}\log u(0,t). We investigate what happens when \kappa\Delta is replaced by \Delta^\cK, where \cK = \{\mathcal{K}(x,y)\colon\,x,y\in\Z^d,\,x \sim y\} is a collection of random conductances between neighbouring sites replacing the constant conductances \kappa in the homogeneous model. We show that the associated annealed Lyapunov exponents are given by the formula \lambda_p(\cK) = \sup\{\lambda_p(\kappa) \colon\,\kappa\in\Supp(\cK)\}, where \Supp(\cK) is the set of values taken by the \cK-field. We also show that for the associated quenched Lyapunov exponent this formula only provides a lower bound. Our proof is valid for three classes of reversible \xi, and for all \cK satisfying a certain clustering property, namely, there are arbitrarily large balls where \cK is almost constant and close to any value in \Supp(\cK). What our result says is that the Lyapunov exponents are controlled by those pockets of \cK where the conductances are close to the value that maximises the growth in the homogeneous setting.

Collisions of random walks in reversible random graphs

We prove that in any recurrent reversible random rooted graph, two independent simple random walks started at the same vertex collide infinitely often almost surely. This applies to the Uniform Infinite Planar Triangulation and Quadrangulation and to the Incipient Infinite Cluster in $\mathbb{Z}^2$.

The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent

Dans cet article on étudie l’équation parabolique d’Anderson $\partial u(x,t)/\partial t=\kappa\varDelta u(x,t)+\xi(x,t)u(x,t)$, $x\in\mathbb{Z}^{d}$, $t\geq0$, où les champs $u$ et $\xi$ sont à valeurs dans $\mathbb{R}$, $\kappa\in[0,\infty)$ est la constante de diffusion, et $\varDelta $ est le laplacien discret. Le champ $\xi$ joue le rôle d’environnement aléatoire dynamique et dirige l’équation. La condition initiale $u(x,0)=u_{0}(x)$, $x\in\mathbb{Z}^{d}$, est choisie positive et bornée. La solution de l’équation parabolique d’Anderson décrit l’évolution d’un champ de particules effectuant des marches aléatoires simples avec un branchement binaire : les particules sautent au taux $2d\kappa$, se divisent en deux au taux $\xi\vee0$, et meurent au taux $(-\xi)\vee0$. Notre but est de prouver un certain nombre de propriétés basiques de la solution $u$ sous des conditions sur $\xi$ qui sont aussi faibles que possible. Ces propriétés vont servir d’impulsion pour de futur améliorations. Tout au long de cet article nous supposons que $\xi$ est stationnaire et ergodique sous les translations en espace et en temps, n’est pas constant et satisfait $\mathbb{E}(|\xi(0,0)|)<\infty$, où $\mathbb{E}$ représente l’espérance par rapport à $\xi$. Sous une hypothèse très faible sur les queues de la distribution de $\xi$, nous montrons que la solution de l’équation parabolique d’Anderson existe et est unique pour tout $\kappa\in[0,\infty)$. Notre principal objet d’intérêt est l’exposant de Lyapunov quenched $\lambda_{0}(\kappa)=\lim_{t\to\infty}\frac{1}{t}\log u(0,t)$. Il a été prouvé dans Gärtner, den Hollander et Maillard (In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner (2012) 159–193 Springer) que cet exposant existe et est constant $\xi$-a.s., satisfait $\lambda_{0}(0)=\mathbb{E}(\xi(0,0))$ et $\lambda_{0}(\kappa)>\mathbb{E}(\xi(0,0))$ pour $\kappa\in(0,\infty)$, et est tel que $\kappa\mapsto\lambda_{0}(\kappa)$ est globalement lipschitzienne sur $(0,\infty)$ à l’extérieur de n’importe quel voisinage de $0$ où il est fini. Sous certaines conditions faibles de mélange en espace-temps sur $\xi$, nous montrons les propriétés suivantes : (1) $\lambda_{0}(\kappa)$ ne dépend pas de la condition initiale $u_{0}$; (2) $\lambda_{0}(\kappa)<\infty$ pour tout $\kappa\in[0,\infty)$; (3) $\kappa\mapsto\lambda_{0}(\kappa)$ est continue sur $[0,\infty)$ mais pas lipschitzienne en $0$. Nous conjecturons en outre : (4) $\lim_{\kappa\to\infty}[\lambda_{p}(\kappa)-\lambda_{0}(\kappa)]=0$ pour tout $p\in\mathbb{N}$, où $\lambda_{p}(\kappa)=\lim_{t\to\infty}\frac{1}{pt}\log\mathbb{E}([u(0,t)]^{p})$ est le $p$-ième exposant de Lyapunov annealed. (Dans (In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner (2012) 159–193 Springer) les propriétés (1), (2) et (4) n’ont pas été abordées, tandis que la propriété (3) a été prouvée sous des hypothèses beaucoup plus restrictives sur $\xi$.) Finalement, nous prouvons que nos conditions faibles de mélange en espace-temps sur $\xi$ sont satisfaites par plusieurs systèmes de particules en interaction.

Phase transition for finite-speed detection among moving particles

Consider the model where particles are initially distributed on Zd,d≥2, according to a Poisson point process of intensity λ>0, and are moving in continuous time as independent simple symmetric random walks. We study the escape versus detection problem, in which the target, initially placed at the origin of Zd,d≥2, and changing its location on the lattice in time according to some rule, is said to be detected if at some finite time its position coincides with the position of a particle. For any given S>0, we consider the case where the target can move with speed at most S, according to any continuous function and can adapt its motion based on the location of the particles. We show that, for any S>0, there exists a sufficiently small λ∗>0, so that if the initial density of particles λ<λ∗, then the target can avoid detection forever.

The parabolic Anderson model in a dynamic random environment: space-time ergodicity for the quenched Lyapunov exponent

We continue our study of the parabolic Anderson equation @u(x;t)=@t = u(x;t) + (x;t)u(x;t), x2 Z d , t 0, where 2 [0;1) is the diusion constant, is the discrete Laplacian, and plays the role of a dynamic random environment that drives the equation. The initial condition u(x; 0) = u0(x), x2 Z d , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a eld of particles performing independent simple random walks with binary branching: particles jump at rate 2d , split into two at rate _ 0, and die at rate ( )_ 0. We assume that is stationary and ergodic under translations in space and time, is not constant and satises E(j (0; 0)j) E( (0; 0)) for 2 (0;1). In the present paper we show that lim !1 0( ) = E( (0; 0)) under an additional @ ;

Random walk in a high density dynamic random environment

The goal of this note is to prove a law of large numbers for the empirical speed of a green particle that performs a random walk on top of a field of red particles which themselves perform independent simple random walks on Zd, d≥1. The red particles jump at rate 1 and are in a Poisson equilibrium with density μ. The green particle also jumps at rate 1, but uses different transition kernels p′ and p″ depending on whether it sees a red particle or not. It is shown that, in the limit as μ→∞, the speed of the green particle tends to the average jump under p′. This result is far from surprising, but it is non-trivial to prove. The proof that is given in this note is based on techniques that were developed in Kesten and Sidoravicius (2005) to deal with spread-of-infection models. The main difficulty is that, due to particle conservation, space–time correlations in the field of red particles decay slowly. This places the problem in a class of random walks in dynamic random environments for which scaling laws are hard to obtain.

Two-sided random walks conditioned to have no intersections

Let $S^{1},S^{2}$ be independent simple random walks in $\mathbb{Z}^{d}$ ($d=2,3$) started at the origin. We construct two-sided random walk paths conditioned that $S^{1}[0,\infty ) \cap S^{2}[1, \infty ) = \emptyset$ by showing the existence of the following limit:\begin{equation*}\lim _{n \rightarrow \infty } P ( \cdot | S^{1}[0, \tau ^{1} ( n) ] \cap S^{2}[1, \tau ^{2}(n) ] = \emptyset ),\end{equation*}where $\tau^{i}(n) = \inf \{ k \ge 0 : |S^{i} (k) | \ge n \}$. Moreover, we give upper bounds of the rate of the convergence. These are discrete analogues of results for Brownian motion obtained by Lawler.

The Liouville and the intersection properties are equivalent for planar graphs

It is shown that if a planar graph admits no non-constant bounded harmonic functions then the trajectories of two independent simple random walks intersect almost surely.

Some Sufficient Conditions for Infinite Collisions of Simple Random Walks on a Wedge Comb

In this paper, we give some sufficient conditions for the infinite collisions of independent simple random walks on a wedge comb with profile $\{f(n), n\in \ZZ\}$. One interesting result is that if $f(n)$ has a growth order as $n\log n$, then two independent simple random walks on the wedge comb will collide infinitely many times. Another is that if $\{f(n); n\in \ZZ\}$ are given by i.i.d. non-negative random variables with finite mean, then for almost all wedge comb with such profile, three independent simple random walks on it will collide infinitely many times.

Intermittency on catalysts: Voter model

In this paper we study intermittency for the parabolic Anderson equation $\partial u/\partial t=\kappa\Delta u+\gamma\xi u$ with $u:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$, where $\kappa\in[0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, $\gamma\in(0,\infty)$ is the coupling constant, and $\xi:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$ is a space--time random medium. The solution of this equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $\xi$. We focus on the case where $\xi$ is the voter model with opinions 0 and 1 that are updated according to a random walk transition kernel, starting from either the Bernoulli measure $\nu_{\rho}$ or the equilibrium measure $\mu_{\rho}$, where $\rho\in(0,1)$ is the density of 1's. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of $u$. We show that if the random walk transition kernel has zero mean and finite variance, then these exponents are trivial for $1\leq d\leq4$, but display an interesting dependence on the diffusion constant $\kappa$ for $d\geq 5$, with qualitatively different behavior in different dimensions. In earlier work we considered the case where $\xi$ is a field of independent simple random walks in a Poisson equilibrium, respectively, a symmetric exclusion process in a Bernoulli equilibrium, which are both reversible dynamics. In the present work a main obstacle is the nonreversibility of the voter model dynamics, since this precludes the application of spectral techniques. The duality with coalescing random walks is key to our analysis, and leads to a representation formula for the Lyapunov exponents that allows for the application of large deviation estimates.