Tom Hutchcroft, Yuval Peres
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 $\mathbb{Z}^2$.
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This paper concerns the long-term behaviour of a system of interacting random walks labeled by vertices of a finite graph. The model is reversible which allows to use the method of electric networks in the study. In addition, examples of alternative proofs not requiring reversibility are provided.
We study a large class of long-range random walks which take values on the vertices of an N dimensional hypercube. These processes are connected with multivariate Bernoulli autoregression.
The Central Limit Theorem for the random walk on a stationary random network of conductances has been studied by several authors. In one dimension, when conductances and resistances are integrable, and following a method of martingale introduced by S. Kozlov (1985), we can prove the Quenched Central Limit Theorem. In that case the variance of the limit law is not null. When resistances are not integrable, the Annealed Central Limit Theorem with null variance was established by Y. Derriennic and M. Lin (personal communication). The quenched version of this last theorem is proved here, by using …
In this short note, we prove that $v(-ε)=-v(ε)$. Here, $v(ε)$ is the speed of a one-dimensional random walk in a dynamic \emph{reversible} random environment, that jumps to the right (resp. to the left) with probability $1/2+ε$ (resp. $1/2-ε$) if it stands on an occupied site, and vice-versa on an empty site. We work in any setting where $v(ε), v(-ε)$ are well-defined, i.e. a weak LLN holds. The proof relies on a simple coupling argument that holds only in the discrete setting.
Let $Ξ$ be an open and bounded subset of $\bb R^d$, and let $F:Ξ\to\bb R$ be a twice continuously differentiable function. Denote by $Ξ_N$ th discretization of $Ξ$, $Ξ_N = Ξ\cap (N^{-1} \bb Z^d)$, and denote by $X_N(t)$ the continuous-time, nearest-neighbor, random walk on $Ξ_N$ which jumps from $\bs x$ to $\bs y$ at rate $ e^{-(1/2) N [F(\bs y) - F(\bs x)]}$. We examine in this article the metastable behavior of $X_N(t)$ among the wells of the potential $F$.
Correction to "The stepping stone model in a random environment" -- limit theorems for the occupation time and intersection time of reversible random walks in random environments
修正了原工作中关于两个独立随机游走在随机环境中相交局部时间的常数错误，利用可逆随机游走占据时间的新标度极限更新了常数γ。
Particle detection among Random Walks as a non-reversible Random Interlacements Process
本文研究Zd上泊松分布的陷阱粒子对可移动目标粒子的检测问题，通过将问题转化为随机交织渗流框架，证明了次临界u下目标可停留在空位无限簇内逃逸，而u足够大时无法逃逸，并首次将随机交织形式体系扩展到非可逆马尔可夫链。
We present results relating mixing times to the intersection time of branching random walk (BRW) in which the logarithm of the expected number of particles grows at rate of the spectral-gap $\mathrm{gap}$ . This is a finite state space analog of a critical branching process. Namely, we show that the maximal expected hitting time of a state by such a BRW is up to a universal constant larger than the $L_{\infty}$ mixing-time, whereas under transitivity the same is true for the intersection time of two independent such BRWs. Using the same methodology, we show that for a sequence of reversible Ma…
We study the condensation regime of the finite reversible inclusion process, i.e., the inclusion process on a finite graph $S$ with an underlying random walk that admits a reversible measure. We assume that the random walk kernel is irreducible and its reversible measure takes maximum value on a subset of vertices $S_\star\subset S$. We consider initial conditions corresponding to a single condensate that is localized on one of those vertices and study the metastable (or tunneling) dynamics. We find that, if the random walk restricted to $S_\star$ is irreducible, then there exists a single tim…
The *-Edge-Reinforced Random Walk
本文提出 *-边强化随机游走（*-ERRW），统一并推广了经典 ERRW 与 r-依赖 ERRW，证明其在部分可交换意义下对应随机环境中的随机游走，并给出混合律的显式表达式，从而将 Coppersmith-Diaconis 魔法公式推广到 Yaglom 可逆马尔可夫链混合情形。
We consider non-reversible random walks evolving on a potential field in a bounded domain of $\mathbb{R}^d$. We describe the complete metastable behavior of the random walk among the landscape of valleys, and we derive the Eyring-Kramers formula for the mean transition time from a metastable set to a stable set.
We introduce via perturbation a class of random walks in reversible dynamic environments having a spectral gap. In this setting one can apply the mathematical results derived in http://arxiv.org/abs/1602.06322. As first results, we show that the asymptotic velocity is antisymmetric in the perturbative parameter and, for a subclass of random walks, we characterize the velocity and a stationary distribution of the environment seen from the walker as suitable series in the perturbative parameter. We then consider as a special case a random walk on the East model that tends to follow dynamical int…
Let $r: S\times S\to \bb R_+$ be the jump rates of an irreducible random walk on a finite set $S$, reversible with respect to some probability measure $m$. For $α>1$, let $g: \bb N\to \bb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k) = (k/k-1)^α$, $k\ge 2$. Consider a zero range process on $S$ in which a particle jumps from a site $x$, occupied by $k$ particles, to a site $y$ at rate $g(k) r(x,y)$. Let $N$ stand for the total number of particles. In the stationary state, as $N\uparrow\infty$, all particles but a finite number accumulate on one single site. We show in this article that in the time…
We consider a special case of random walk in random environment (RWRE) on Z^d where the environment is periodic (RWPE). Under natural conditions, we show that law of large numbers and central limit theorem holds. In the ballistic nearest neighbour reversible case, we prove that the angle between the asymptotic direction of the RWPE and the average negative gradient of the potential function of the reversible environment is less than pi/2, that is, the potential cannot increase asymptotically along the trajectory of the RWPE. But this angle can be close to pi/2.
We study a random walk in a random environment (RWRE) on $\Z^d$, $1 \leq d < +\infty$. The main assumptions are that conditionned on the environment the random walk is reversible. Moreover we construct our environment in such a way that the walk can't be trapped on a single point like in some particular RWRE but in some specific d-1 surfaces. These surfaces are basic surfaces with deterministic geometry. We prove that the local time in the neighborhood of these surfaces is driven by a function of the (random) reversible measure. As an application we get the limit law of the local time as a pro…
We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.
Consider a system of coalescing random walks where each individual performs random walk over a finite graph G, or (more generally) evolves according to some reversible Markov chain generator Q. Let C be the first time at which all walkers have coalesced into a single cluster. C is closely related to the consensus time of the voter model for this G or Q. We prove that the expected value of C is at most a constant multiple of the largest hitting time of an element in the state space. This solves a problem posed by Aldous and Fill and gives sharp bounds in many examples, including all vertex-tran…
Suppose X and Y are two independent irreducible Markov chains on n states. We consider the intersection time, which is the first time their trajectories intersect. We show for reversible and lazy chains that the total variation mixing time is always upper bounded by the expected intersection time taken over the worst starting states. For random walks on trees we show the two quantities are equivalent. We obtain an expression for the expected intersection time in terms of the eigenvalues for reversible and transitive chains. For such chains we also show that it is up to constants the geometric …
We study the persistence probability for some discrete-time, time-reversible processes. In particular, we deduce the persistence exponent in a number of examples: first, we deal with random walks in random sceneries (RWRS) in any dimension with Gaussian scenery. Second, we deal with sums of stationary Gaussian sequences with correlations exhibiting long-range dependence. Apart from the persistence probability we deal with the position of the maximum and the time spent on the positive half-axis by the process.
We study dynamic random conductance models on $\mathbb{Z}^2$ in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walks on dynamical percolation.
It is shown that transient graphs for the simple random walk do not admit a nearest neighbor transient Markov chain (not necessarily a reversible one), that crosses all edges with positive probability, while there is such chain for the square grid $\mathbb{Z}^2$. In particular, the $d$-dimensional grid $\mathbb{Z}^d$ admits such a Markov chain only when $d=2$. For $d=2$ we present a relevant example due to Gady Kozma, while the general statement for transient graphs is obtained by proving that for every transient irreducible Markov chain on a countable state space, which admits a stationary me…
Collision properties of the four-dimensional random walk trace
研究四维整数格点上简单随机游走迹线上多个随机游走的碰撞，对两个独立连续时间游走得到碰撞时间过程的标度极限，对三个独立离散时间游走证明存在无穷多次三重碰撞。
Quenched local limit theorem for random conductance models with long-range jumps
该研究针对带任意长度跳跃的平稳遍历随机电导模型，利用弱抛物Harnack不等式与对角热核估计，建立了d≥2维整数格点上可逆随机游走的淬火局部极限定理，部分解决了相关开放问题，还得到带额外尾项的极大不等式与校正子的处处次线性性质。
We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger, $L^{2+\varepsilon}$ (rather than $L^2$) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.
The constant speed random walk on dynamical conductances
本文研究动态随机电导中的恒速有偏随机游走，证明其在特定参数范围内为弹道式，并在全参数范围内建立退火与淬火中心极限定理，同时指出不可逆性导致经典爱因斯坦关系失效。
We construct a class of non-reversible Metropolis kernels as a multivariate extension of the guided-walk kernel proposed by Gustafson 1998. The main idea of our method is to introduce a projection that maps a state space to a totally ordered group. By using Haar measure, we construct a novel Markov kernel termed Haar-mixture kernel, which is of interest in its own right. This is achieved by inducing a topological structure to the totally ordered group. Our proposed method, the Delta-guided Metropolis--Haar kernel, is constructed by using the Haar-mixture kernel as a proposal kernel. The propos…
These notes cover one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. The aim is to review recent mathematical developments in the field of random walks in random environment. Our main focus will be on directionally transient and reversible random walks on different types of underlying graph structures, such as $\mathbb{Z}$, trees and $\mathbb{Z}^d$ for $d\geq 2$.
Poissonization-based collision threshold derivation for random walks on lattices
本文利用泊松化技术将格上两个独立简单随机游走的碰撞概率转化为连续时间差分游走的返回概率，并通过修正Bessel函数的渐近分析证明期望碰撞次数有限当且仅当维数d≥3，同时给出渐近碰撞次数的一般公式。
We consider a weighted lattice $Z^d$ with conductance $μ_e=|e|^{-α}$. We show that the heat kernel of a variable speed random walk on it satisfies a two-sided Gaussian bound by using an intrinsic metric. We also show that when $d=2$ and $α\in (-1,0)$, two independent random walks on such weighted lattice will collide infinite many times while they are transient.
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.
Let $P$ be the transition matrix of a finite, irreducible and reversible Markov chain. We say the continuous time Markov chain $X$ has transition matrix $P$ and speed $λ$ if it jumps at rate $λ$ according to the matrix $P$. Fix $λ_X,λ_Y,λ_Z\geq 0$, then let $X,Y$ and $Z$ be independent Markov chains with transition matrix $P$ and speeds $λ_X,λ_Y$ and $λ_Z$ respectively, all started from the stationary distribution. What is the chance that $X$ and $Y$ meet before either of them collides with $Z$? For each choice of $λ_X,λ_Y$ and $λ_Z$ with $\max(λ_X,λ_Y)>0$, we prove a lower bound for this prob…
We prove that for recurrent, reversible graphs, the following conditions are equivalent: (a) existence and uniqueness of the potential kernel, (b) existence and uniqueness of harmonic measure from infinity, (c) a new anchored Harnack inequality, and (d) one-endedness of the wired Uniform Spanning Tree. In particular this gives a proof of the anchored (and in fact also elliptic) Harnack inequality on the UIPT. This also complements and strengthens some results of Benjamini, Lyons, Peres and Schramm. Furthermore, we make progress towards a conjecture of Aldous and Lyons by proving that these con…
Cutoff for mixtures of permuted Markov chains: reversible case
研究了随机环境中可逆马尔可夫链的混合性质，发现其在熵时间log n/h处出现截止现象，且证明了在弱假设下高概率成立，同时提出了一种新的关于对称群上低度函数的集中性结果。
Consider a probability measure on a Hilbert space defined via its density with respect to a Gaussian. The purpose of this paper is to demonstrate that an appropriately defined Markov chain, which is reversible with respect to the measure in question, exhibits a diffusion limit to a noisy gradient flow, also reversible with respect to the same measure. The Markov chain is defined by applying a Metropolis-Hastings accept-reject mechanism to an Ornstein-Uhlenbeck proposal which is itself reversible with respect to the underlying Gaussian measure. The resulting noisy gradient flow is a stochastic …
A recurrent graph $G$ has the infinite collision property if two independent random walks on $G$, started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton-Watson tree with finite variance conditioned to survive, the incipient infinite cluster in $\Z^d$ with $d \ge 19$ and the uniform spanning tree in $\Z^2$ all have the infinite collision property. For power-law combs and spherically symmetric trees, we determine precisely the phase boundary for the infinite coll…
Scaling limit of the collision measure for two-dimensional random walks
研究二维随机游走碰撞测度的对数尺度极限，证明其依分布收敛到由泊松点过程表示的随机测度，并通过簇分解揭示碰撞结构。
Collisions of random walks in unimodular random graphs: applications to the random geometric graph and long-range percolation
研究单模随机根图中简单随机游走的碰撞性质，在常返及根的可积条件下两独立随机游走几乎必然无限次碰撞，瞬态及相关可积假设下几乎必然仅有限次碰撞，还应用于多种无界度随机图及长程渗流模型。
For an integer $k\ge 2$, let $S^{(1)}, S^{(2)}, \dots, S^{(k)}$ be $k$ independent simple symmetric random walks on $\mathbb{Z}$. A pair $(n,z)$ is called a collision event if there are at least two distinct random walks, namely, $S^{(i)},S^{(j)}$ satisfying $S^{(i)}_n= S^{(j)}_n=z$. We show that under the same scaling as in Donsker's theorem, the sequence of random measures representing these collision events converges to a non-trivial random measure on $[0,1]\times \mathbb{R}$. Moreover, the limit random measure can be characterized using Wiener chaos. The proof is inspired by methods from s…
We derive a quenched invariance principle for random walks in random environments whose transition probabilities are defined in terms of weighted cycles of bounded length. To this end, we adapt the proof for random walks among random conductances by Sidoravicius and Sznitman (Probab. Theory Related Fields 129 (2004) 219--244) to the non-reversible setting.
A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees an…
We consider d independent walkers on Z, m of them performing simple symmetric random walk and r = d -- m of them performing recurrent RWRE (Sinai walk), in I independent random environments. We show that the product is recurrent, almost surely, if and only if m $\le$ 1 or m = d = 2. In the transient case with r $\ge$ 1, we prove that the walkers meet infinitely often, almost surely, if and only if m = 2 and r $\ge$ I = 1. In particular, while I does not have an influence for the recurrence or transience, it does play a role for the probability to have infinitely many meetings. To obtain these …
Attributing a positive value τ_x to each x in Z^d, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (τ_x), often known as "Bouchaud's trap model". We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d > 4. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as a time change of a random walk among random conductances. We then focus on proving that the tim…
We study asymptotic limits of reversible random walks on tessellations via a variational approach, which relies on a specific generalized-gradient-flow formulation of the corresponding forward Kolmogorov equation. We establish sufficient conditions on sequences of tessellations and jump intensities under which a sequence of random walks converges to a diffusion process with a possibly spatially-dependent diffusion tensor.
We give a lower bound for the non-collision probability up to a long time T in a system of n independent random walks with fixed obstacles on the two-dimensional lattice. By `collision' we mean collision between the random walks as well as collision with the fixed obstacles. We give an analogous result for Brownian particles on the plane. We also explain how this result can be used to describe in terms of "quasi random walks" a diluted gas evolving under Kawasaki dynamics or simple exclusion.
Infinite collisions of simple random walks on random recursive trees generated by Bernoulli sequences
研究伯努利序列生成的随机递归树，通过从特定图递归构建，证明其几乎必然恰有一个拓扑端，且具有无限碰撞性质，即两个独立简单随机游走几乎必然无限次碰撞。
We study the spectral theory of a reversible Markov chain associated to a hypoelliptic random walk on a manifold M. This random walk depends on a parameter h which is roughly the size of each step of the walk. We prove uniform bounds with respect to h on the rate of convergence to equilibrium, and the convergence when h goes to zero to the associated hypoelliptic diffusion.
This work is motivated by the study of some two-dimensional random walks in random environment (RWRE) with transition probabilities independent of one coordinate of the walk. These are non-reversible models and can not be treated by electrical network techniques. The proof of the recurrence of such RWRE needs new estimates for quenched return probabilities of a one-dimensional recurrent RWRE. We obtained these estimates by constructing suitable valleys for the potential. They imply that k independent walkers in the same one-dimensional (recurrent) environment will meet in the origin infinitely…
Let $G$ be a connected graph of uniformly bounded degree. A $k$ non-backtracking random walk ($k$-NBRW) $(X_n)_{n =0}^{\infty}$ on $G$ evolves according to the following rule: Given $ (X_n)_{n =0}^{s}$, at time $s+1$ the walk picks at random some edge which is incident to $X_s$ that was not crossed in the last $k$ steps and moves to its other end-point. If no such edge exists then it makes a simple random walk step. Assume that for some $R>0$ every ball of radius $R$ in $G$ contains a simple cycle of length at least $k$. We show that under some "nice" random time change the $k$-NBRW becomes re…
We provide conditions that classify cover times for sequences of random walks on random graphs into two types: One type (Type 1) is the class of cover times that are of the order of the maximal hitting times scaled by the logarithm of the size of vertex sets. The other type (Type 2) is the class of cover times that are of the order of the maximal hitting times. The conditions are described by some parameters determined by the underlying graphs: the volumes, the diameters with respect to the resistance metric, the coverings or packings by balls in the resistance metric. We apply the conditions …
Metastable behavior of weakly mixing Markov chains: the case of reversible, critical zero-range processes
本文提出基于预解式方程的弱混合马尔可夫链亚稳态行为通用推导方法，突破现有混合条件限制，并将其应用于临界零程过程，证明凝聚位点在$N^2 \log N$时间尺度上按与底层游走容量成正比的转移速率演化。
Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^3$, whose probability law is denoted by $\mathbf{P}$. For $\mathbf{P}$-a.s. realization of $\mathcal{U}$, the recurrence of the the simple random walk on $\mathcal{U}$ is proved in [5] and it is also demonstrated in [8] that two independent simple random walks on $\mathcal{U}$ collide infinitely often. In this article, we will give a quantitative estimate on the number of collisions of two independent simple random walks on $\mathcal{U}$, which provides another proof of the infinite collision property of $\mathcal{U}$.
We prove tail estimates for variables $\sum_i f(X_i)$, where $(X_i)_i$ is the trajectory of a random walk on an undirected graph (or, equivalently, a reversible Markov chain). The estimates are in terms of the maximum of the function $f$, its variance, and the spectrum of the graph. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein and Bennett-type inequalities, as well as an inequality for subgaussian variables.
We consider a random walk on a random graph $(V,E)$, where $V$ is the set of open sites under i.i.d. Bernoulli site percolation on the multi-dimensional integer set $\mathbf{Z}^d$, and the transition probabilities of the walk are generated by i.i.d. random conductances (positive numbers) assigned to the edges in $E$. This random walk in random environments has long range jumps and is reversible. We prove the quenched invariance principle for this walk when the random conductances are unbounded from above but uniformly bounded from zero by taking the corrector approach. To this end, we prove a …
We consider the random conductance model, where the underlying graph is an infinite supercritical Galton--Watson tree, the conductances are independent but their distribution may depend on the degree of the incident vertices. We prove that, if the mean conductance is finite, there is a deterministic, strictly positive speed $v$ such that $\lim_{n\to\infty} \frac{|X_n|}{n}= v$ a.s.\ (here, $|\cdot|$ stands for the distance from the root). We give a formula for $v$ in terms of the laws of certain effective conductances and show that, if the conductances share the same expected value, the speed i…
Limit Profiles for Reversible Markov Chains
本文将 Teyssier 的极限剖面逼近方法从群上共轭不变随机游走推广到齐性空间随机游走和一般可逆 Markov 链，并用于改进 k-循环洗牌、Ehrenfest 瓮扩散及 Gibbs 采样器等经典问题的结果。
Collisions of random walks on comb graphs with a planar base
本文研究以递归平面图为基底的梳状图上两个独立随机游走的碰撞次数，证明其随齿长轮廓函数的增长发生相变，并解决了 Barlow 等人关于 $\mathbb{Z}^2$ 基底上对数增长轮廓的碰撞有限性与无限性的临界问题。
Infinitely many collisions between a recurrent simple random walk and arbitrary many transient random walks in a subballistic random environment
本文证明在亚弹道随机环境中，一个常返简单随机游走与任意多个暂态随机游走几乎必然在无穷多个时刻同时碰撞于同一位置。
Let $Γ$ be a graph and $P$ be a reversible random walk on $Γ$. From the $L^2$ analyticity of the Markov operator $P$, we deduce that an iterate of odd exponent of $P$ is `lazy', that is there exists an integer $k$ such that the transition probability (for the random walk $P^{2k+1}$) from a vertex $x$ to itself is uniformly bounded from below. The proof does not require the doubling property on $Γ$ but only a polynomial control of the volume.
We show bounds on total variation and $L^{\infty}$ mixing times, spectral gap and magnitudes of the complex valued eigenvalues of a general (non-reversible non-lazy) Markov chain with a minor expansion property. This leads to the first known bounds for the non-lazy simple and max-degree walks on a (directed) graph, and even in the lazy case they are the first bounds of the optimal order. In particular, it is found that within a factor of two or four, the worst case of each of these mixing time and eigenvalue quantities is a walk on a cycle with clockwise drift.
In this paper we study the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) introduced in [4] by continuous time random walks on square lattices. The state space of BMVD contains a $2$-dimensional component, a $3$-dimensional component, and a "darning point" which joins these two components. Such a state space is equipped with the geodesic distance, under which BMVD is a diffusion process. In this paper, we prove that BMVD restricted on a bounded domain containing the darning point is the weak limit of continuous time reversible random walks with exponent…
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