Recurrence of random walk traces

平面大象随机游走的常返性

Recurrence of the plane Elephant random walk

本文通过与标准平面随机游走进行粗略比较，给出了扩散状态下二维大象随机游走常返性的简短证明，并期望该方法可推广至其他问题。

Quantitative recurrence results for random walks

First, we prove a \emph{local almost sure central limit theorem} for lattice random walks in the plane. The corresponding version for random walks in the line was considered by the author in \cite{5}. This gives us a quantitative version of Pólya's Recurrence Theorem \cite{6}. Second, we prove a \emph{local almost sure central limit theorem} for (not necessarly lattice) random walks in the line or in the plane, which will also give us quantitative recurrence results. Finally, we prove an \emph{almost sure central limit theorem} for multidimensional (not necessarly lattice) random walks. This i…

Recurrence for vertex-reinforced random walks on Z with weak reinforcements

We prove that any vertex-reinforced random walk on the integer lattice with non-decreasing reinforcement sequence $w$ satisfying $w(k) = o(k^α)$ for some $α< 1/2$ is recurrent. This improves on previous results of Volkov (2006) and Schapira (2012).

Recurrence and transience of excited random walks on $\Z^d$ and strips

We investigate excited random walks on $\Z^d, d\ge 1,$ and on planar strips $\Z\times\{0,1,...,L-1\}$ which have a drift in a given direction. The strength of the drift may depend on a random i.i.d. environment and on the local time of the walk. We give exact criteria for recurrence and transience, thus generalizing results by Benjamini and Wilson for once-excited random walk on $\Z^d$ and by the author for multi-excited random walk on $\Z$.

Degrees of transience and recurrence and hierarchical random walks

The notion of degree and related notions concerning recurrence and transience for a class of L'evy processes on metric Abelian groups are studied. The case of random walks on a hierarchical group is examined with emphasis on the role of the ultrametric structure of the group and on analogies and differences with Euclidean random walks. Applications to separation of time scales and occupation times of multilevel branching systems are discussed.

On recurrence of reflected random walk on the half-line. With an appendix on results of Martin Benda

Let $(Y_n)$ be a sequence of i.i.d. real valued random variables. Reflected random walk $(X_n)$ is defined recursively by $X_0=x \ge 0$, $X_{n+1} = |X_n - Y_{n+1}|$. In this note, we study recurrence of this process, extending a previous criterion. This is obtained by determining an invariant measure of the embedded process of reflections.

On the recurrence set of planar Markov Random Walks

In this paper, we investigate the properties of recurrent planar Markov random walks. More precisely, we study the set of recurrent points with the use of local limit theorems. The Nagaev-Guivarc'h spectral method provides several examples for which these local limit theorems are satisfied as soon as the (standard or non-standard) central limit theorem holds.

Recurrence of Multidimensional Persistent Random Walks. Fourier and Series Criteria

The recurrence features of persistent random walks built from variable length Markov chains are investigated. We observe that these stochastic processes can be seen as L{é}vy walks for which the persistence times depend on some internal Markov chain: they admit Markov random walk skeletons. A recurrence versus transience dichotomy is highlighted. We first give a sufficient Fourier criterion for the recurrence, close to the usual Chung-Fuchs one, assuming in addition the positive recurrence of the driving chain and a series criterion is derived. The key tool is the Nagaev-Guivarc'h method. Fina…

Recurrence and Transience for Branching Random Walks in an iid Random Environment

We give three different criteria for transience of a Branching Markov Chain. These conditions enable us to give a classification of Branching Random Walks in Random Environment (BRWRE) on Cayley Graphs in recurrence and transience. This classification is stated explicitly for BRWRE on $\Z^d.$ Furthermore, we emphasize the interplay between Branching Markov Chains and the spectral radius. We prove properties of the spectral radius of the Random Walk in Random Environment with the help of appropriate Branching Markov Chains.

Recurrence of a Weighted Random Walk on a Circle Packing with Parabolic Carrier

In this paper we show that given a circle packing of an infinite planar triangulation such that its carrier is parabolic, placing weights on the edges according to a certain natural way introduced by Dubejko, makes the random walk recurrent. We also propose a higher-dimensional analogue of the Dubejko weights.

Recurrence and transience of a multi-excited random walk on a regular tree

We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition of the recurrence/transience property of the walk. In particular, we prove that the asymptotic behavior of the walk depends on the order of the excitations, which contrasts with the one dimensional setting studied by Zerner (2005). We also consider the limiting speed of the walk in the transient regime and conjecture that it is not a monotonic function of the environment.

Recurrence properties of a special type of Heavy-Tailed Random Walk

In the proof of the invariance principle for locally perturbed periodic Lorentz process with finite horizon, a lot of delicate results were needed concerning the recurrence properties of its unperturbed version. These were analogous to the similar properties of Simple Symmetric Random Walk. However, in the case of Lorentz process with infinite horizon, the analogous results for the corresponding random walk are not known, either. In this paper, these properties are ascertained for the appropriate random walk (this happens to be in the non normal domain of attraction of the normal law). As a to…

Recurrence of Simple Random Walk on $Z^2$ is Dynamically Sensitive

Benjamini, Haggstrom, Peres and Steif introduced the concept of a dynamical random walk. This is a continuous family of random walks, {S_n(t)}. Benjamini et. al. proved that if d=3 or d=4 then there is an exceptional set of t such that {S_n(t)} returns to the origin infinitely often. In this paper we consider a dynamical random walk on Z^2. We show that with probability one there exists t such that {S_n(t)} never returns to the origin. This exceptional set of times has dimension one. This proves a conjecture of Benjamini et. al.

Anomalous recurrence properties of many-dimensional zero-drift random walks

Famously, a $d$-dimensional, spatially homogeneous random walk whose increments are non-degenerate, have finite second moments, and have zero mean is recurrent if $d \in \{1,2\}$ but transient if $d \geq 3$. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially non-homogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension $d \geq 2$, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by rand…

Site recurrence for coalescing random walk

Begin continuous time random walks from every vertex of a graph and have particles coalesce when they collide. We use a duality relation with the voter model to prove the process is site recurrent on bounded degree graphs, and for Galton-Watson trees whose offspring distribution has exponential tail. We prove bounds on the occupation probability of a site, as well as a general 0-1 law. Similar conclusions hold for a coalescing process on trees where particles do not backtrack.

Transience/Recurrence and the speed of a one-dimensional random walk in a "have your cookie and eat it" environment

Consider a simple random walk on the integers with the following transition mechanism. At each site $x$, the probability of jumping to the right is $ω(x)\in[\frac12,1)$, until the first time the process jumps to the left from site $x$, from which time onward the probability of jumping to the right is $\frac12$. We investigate the transience/recurrence properties of this process in both deterministic and stationary, ergodic environments $\{ω(x)\}_{x\in Z}$. In deterministic environments, we also study the speed of the process.

On the recurrence of some random walks in random environment

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…

A note on the recurrence of edge reinforced random walks

We give a short proof of Theorem 2.1 from [MR07], stating that the linearly edge reinforced random walk (ERRW) on a locally finite graph is recurrent if and only if it returns to its starting point almost surely. This result was proved in [MR07] by means of the much stronger property that the law of the ERRW is a mixture of Markov chains. Our proof only uses this latter property on finite graphs, in which case it is a consequence of De Finetti's theorem on exchangeability.

Recurrence or transience of random walks on random graphs generated by point processes in $\mathbb{R}^d$

We consider random walks associated with conductances on Delaunay triangulations, Gabriel graphs and skeletons of Voronoi tilings which are generated by point processes in $\mathbb{R}^d$. Under suitable assumptions on point processes and conductances, we show that, for almost any realization of the point process, these random walks are recurrent if $d=2$ and transient if $d\geq 3$. These results hold for a large variety of point processes including Poisson point processes, Mat{é}rn cluster and Mat{é}rn hardcore processes which have clustering or repulsive properties. In order to prove them, we…

Tree Builder Random Walk: recurrence, transience and ballisticity

The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and attached to the current position of the walker. These processes share similarities with other important classes of markovian and non-markovian random walks presenting a large variety of behaviors according to parameters specifications. We show that for a large and most significant class of tree builder random walks, the process is either null recurrent or trans…

Level Crossing Probabilities II: Polygonal Recurrence of Multidimensional Random Walks

In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is of the maximal order square root of n. In higher dimensions we call a random walk 'polygonally recurrent' (resp. transient) if a.s. infinitely many (resp. finitely many) of the straight lines between two consecutive sites hit a given bounded set. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z^3 in the…

Recurrence of edge-reinforced random walk on a two-dimensional graph

We consider a linearly edge-reinforced random walk on a class of two-dimensional graphs with constant initial weights. The graphs are obtained from $\mathbb{Z}^2$ by replacing every edge by a sufficiently large, but fixed number of edges in series. We prove that the linearly edge-reinforced random walk on these graphs is recurrent. Furthermore, we derive bounds for the probability that the edge-reinforced random walk hits the boundary of a large box before returning to its starting point.

平衡激发随机游走 M(2,1,2) 的常返性与访问范围

Recurrence and range of the balanced excited random walk M(2,1,2)

本文证明平面平衡激发随机游走 M(2,1,2) 常返，其访问范围极限与平面简单随机游走相同，并推广至空间非均匀饼干环境。

生长树与超立方体上随机游走的常返/暂留性分析

An Analysis of the Recurrence/Transience of Random Walks on Growing Trees and Hypercubes

该研究针对生长图上随机游走（RWoGG）模型，证明了k叉生长树、无限增长n维超立方体等图上的随机游走的常返性与暂留性会随图生长速度发生相变，还引入LHaGG概念通过耦合论证完成证明。

时间非齐次随机游走的常返与暂态条件

Conditions for recurrence and transience for time-inhomogeneous random walks

本文扩展了Abramov关于时间非齐次生灭过程的结果至时间非齐次随机游走，完整解决了Menshikov和Volkov提出的开放问题。

Random walk in cooling random environment: recurrence versus transience and mixed fluctuations

This is the third in a series of papers in which we consider one-dimensional Random Walk in Cooling Random Environment (RWCRE). The latter is obtained by starting from one-dimensional Random Walk in Random Environment (RWRE) and resampling the environment along a sequence of deterministic times, called refreshing times. In the present paper we explore two questions for general refreshing times. First, we investigate how the recurrence versus transience criterion known for RWRE changes for RWCRE. Second, we explore the fluctuations for RWCRE when RWRE is either recurrent or satisfies a classica…

随机电导环境下临界随机游走蛇的常返性与暂留性

Recurrence and transience of the critical random walk snake in random conductances

本文研究临界Galton-Watson树上分支随机游走在随机电导或陷阱环境中的常返性与暂留性，证明在有限三阶矩及适当技术条件下，过程在四维及以下常返、更高维暂留，证明基于截断二阶矩方法。

Recurrence/Transience criteria for excited random walks with finite-drift cookie stacks

We consider excited random walk (ERW) on $\mathbb{Z}$ in environments with identical stacks of infinitely many cookies at each site, subject to the constraint that the total drift per site $δ= \sum (2p_j - 1)$ is finite. Building on the methods of Kozma, Orenshtein, and Shinkar (arXiv:1311.7439), we show that ERW in finite-drift environments is recurrent when $|δ|<1$ and transient when $|δ|>1$. In the case $|δ|=1$ we prove that ERW is recurrent under mild assumptions on the environment. In addition, we show that ERW may be transient when $|δ|=1$, an interesting new behavior that was not presen…

Remarks on the recurrence and transience of non-backtracking random walks

A short proof of the equivalence of the recurrence of non-backtracking random walk and that of simple random walk on regular infinite graphs is given. It is then shown how this proof can be extended in certain cases where the graph in question is not regular.

Recurrence rates and hitting-time distributions for random walks on the line

We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as r goes to zero, of the return time to (-r,r). We then refine this result by establishing a limit theorem for the hitting-time distributions of (x-r,x+r) with arbitrary real x.

Recurrence of 2-dimensional queueing processes, and random walk exit times from the quadrant

Let $X = (X_1, X_2)$ be a 2-dimensional random variable and $X(n), n \in \mathbb{N}$ a sequence of i.i.d. copies of $X$. The associated random walk is $S(n)= X(1) + \cdots +X(n)$. The corresponding absorbed-reflected walk $W(n), n \in \mathbb{N}$ in the first quadrant is given by $W(0) = x \in \mathbb{R}_+^2$ and $W(n) = \max \{ 0, W(n-1) - X(n) \}$, where the maximum is taken coordinate-wise. This is often called the Lindley process and models the waiting times in a two-server queue. We characterize recurrence of this process, assuming suitable, rather mild moment conditions on $X$. It turns …

On Mixing Behavior of a Family of Random Walks Determined by a Linear Recurrence

We study random walks on the integers mod $G_n$ that are determined by an integer sequence $\{ G_n \}_{n \geq 1}$ generated by a linear recurrence relation. Fourier analysis provides explicit formulas to compute the eigenvalues of the transition matrices and we use this to bound the mixing time of the random walks.

Asymptotic Analysis of the Elephant Random Walk

In this work we study asymptotic properties of a long range memory random walk known as elephant random walk. First we prove recurrence and positive recurrence for the elephant random walk. Then, we establish the transience regime of the model. Finally, under the Poisson Hypothesis, we study the replica mean field limit for this random walk and we obtain an upper bound for the expected distance of the walker from the origin.

Dynamical sensitivity of recurrence and transience of branching random walks

Consider a sequence of i.i.d. random variables $X_n$ where each random variable is refreshed independently according to a Poisson clock. At any fixed time $t$ the law of the sequence is the same as for the sequence at time 0 but at random times almost sure properties of the sequence may be violated. If there are such \emph{exceptional times} we say that the property is \emph{dynamically sensitive}, otherwise we call it \emph{dynamically stable}. In this note we consider branching random walks on Cayley graphs and prove that recurrence and transience are dynamically stable in the sub-and superc…

具有双记忆通道的象随机游走的递归性、暂态性与逃逸速率

Recurrence, Transience, and the Rate of Escape for Elephant Random Walks with Two Memory Channels

研究双记忆通道象随机游走，证明p=11/16为精确递归-暂态阈值，并确定暂态区间的逃逸速率及p=7/8时的逃逸尺度。

多维大象随机游走的常返性与非常返性

Recurrence and transience of multidimensional elephant random walks

本文证明了多维大象随机游走在$d\geq3$时非常返且零点期望有限，在$d=1,2$时常返与非常返相变发生于$p=(2d+1)/(4d)$，并给出了不同$p$参数下的Berry-Esseen型界与极限分布研究。

On multidimensional branching random walks in random environment

We study branching random walks in random i.i.d. environment in $\Z^d, d \geq 1$. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time.

Recurrence for persistent random walks in two dimensions

We discuss the question of recurrence for persistent, or Newtonian, random walks in Z^2, i.e., random walks whose transition probabilities depend both on the walker's position and incoming direction. We use results by Toth and Schmidt-Conze to prove recurrence for a large class of such processes, including all "invertible" walks in elliptic random environments. Furthermore, rewriting our Newtonian walks as ordinary random walks in a suitable graph, we gain a better idea of the geometric features of the problem, and obtain further examples of recurrence.

Almost Sure Recurrence of the Simple Random Walk Path

It is shown that the path of a simple random walk on any graph, consisting of all vertices visited and edges crossed by the walk, is almost surely a recurrent subgraph.

复发、暂留与Rademacher随机游走的反集中性

Recurrence, transience and anti-concentration of Rademacher random walks

本文研究了一维Rademacher随机游走的暂留与复发性，证明了步长序列有界时游走弱递归，步长增长缓慢时可能暂留，但步长快速增长仍可保持弱递归，且当步长增长速度超过n^{1/2}时游走暂留。

Transience, Recurrence and the Speed of a Random Walk in a Site-Based Feedback Environment

We study a random walk on $\mathbb{Z}$ which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, $p$ and $q$. $R$ consecutive right jumps from a site in the $q$-mode are required to switch it to the $p$-mode, and $L$ consecutive left jumps from a site in the $p$-mode are required to switch it to the $q$-mode. From a site in the $p$-mode the walk jumps right with probability $p$ and left with probability $1-p$, while from a site in the $q$-mode these probabilities are $q$ and $1-q$. We prove a sharp cutoff for right/left transience of …

On a class of quantum channels, open random walks and recurrence

We study a particular class of trace-preserving completely positive maps, called PQ-channels, for which classical and quantum evolutions are isolated in a certain sense. By combining open quantum random walks with a notion of recurrence, we are able to describe criteria for recurrence of the walk related to this class of channels. Positive recurrence for open walks is also discussed in this context.

The branching-ruin number and the critical parameter of once-reinforced random walk on trees

The motivation for this paper is the study of the phase transition for recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define a quantity, that we call the branching-ruin number of a tree, which provides (in the spirit of Furstenberg, 1970, and Lyons, 1990) a natural way to measure trees with polynomial growth. We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk. We define a sharp and effective (i.e. co…

Recurrence of cocycles and stationary random walks

We survey distributional properties of $\mathbb{R}^d$-valued cocycles of finite measure preserving ergodic transformations (or, equivalently, of stationary random walks in $\mathbb{R}^d$) which determine recurrence or transience.

Transience and recurrence of random walks on percolation clusters in an ultrametric space

We study existence of percolation in the hierarchical group of order $N$, which is an ultrametric space, and transience and recurrence of random walks on the percolation clusters. The connection probability on the hierarchical group for two points separated by distance $k$ is of the form $c_k/N^{k(1+δ)}, δ>-1$, with $c_k=C_0+C_1\log k+C_2k^α$, non-negative constants $C_0, C_1, C_2$, and $α>0$. Percolation was proved in Dawson and Gorostiza (2013) for $δ<1$, and for the critical case, $δ=1,C_2>0$, with $α>2$. In this paper we improve the result for the critical case by showing percolation for $…

Transience and recurrence of sets for branching random walk via non-standard stochastic orders

We study how the recurrence and transience of space-time sets for a branching random walk on a graph depends on the offspring distribution. Here, we say that a space-time set $A$ is recurrent if it is visited infinitely often almost surely on the event that the branching random walk survives forever, and say that $A$ is transient if it is visited at most finitely often almost surely. We prove that if $μ$ and $ν$ are supercritical offspring distributions with means $\bar μ< \bar ν$ then every space-time set that is recurrent with respect to the offspring distribution $μ$ is also recurrent with …

Central Limit Theorem and recurrence for random walks in bistochastic random environments

We prove the annealed Central Limit Theorem for random walks in bistochastic random environments on $Z^d$ with zero local drift. The proof is based on a "dynamicist's interpretation" of the system, and requires a much weaker condition than the customary uniform ellipticity. Moreover, recurrence is derived for $d \le 2$.

A central limit theorem for random walk in random environment on marked Galton-Watson trees

In this article we focus on a general model of random walk on random marked trees. We prove a recurrence criterion, analogue to the recurrence criterion proved by R. Lyons and Robin Pemantle (1992) in a slightly different model. In the critical case, we obtain a criterion for the positive/null recurrence. Several regimes appear, as proved (in a similar model), by Y. Hu and Z. Shi (2007). We focus on the "diffusive" regime and improve their result in this case, by obtaining a functional Central Limit Theorem. Our result is also an extension of a result by Y. Peres and O. Zeitouni (2008), obtain…

Perturbing transient Random Walk in a Random Environment with cookies of maximal strength

We consider a left-transient random walk in a random environment on Z that will be disturbed by cookies inducing a drift to the right of strength 1. The number of cookies per site is i.i.d. and independent of the environment. Criteria for recurrence and transience of the random walk are obtained. For this purpose we use subcritical branching processes in random environments with immigration and formulate criteria for recurrence and transience for these processes.

Recurrence of the twisted planar random walk

We show that the "twisted" planar random walk - which results by summing up stationary increments rotated by multiples of a fixed angle - is recurrent under diverse assumptions on the increment process. For example, if the increment process is alpha-mixing and of finite second moment, then the twisted random walk is recurrent for every angle fixed choice of the angle out of a set of full Lebesgue measure, no matter how slow the mixing coefficients decay.

慢变化环境中的随机游走

Random walk in slowly changing environments

针对变化环境中随机游走（RWCE）的常返暂留判定难题，提出适用于任意图的慢变化边权有界RWCE的判定条件，证明其常返/暂留性与初始加权图一致。

Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces

We discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular we show that a closed subgroup of a product of finitely many linear groups over local fields supports a recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces. The structure of closed subgroups of linear groups over local fields and the properties of group actions with respect to stationary me…

Recurrence of the frog model on the 3,2-alternating tree

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.

稳定分支随机游走的递归性与容量

Recurrence and capacity of stable branching random walks

本文针对后代分布为临界且吸引α-稳定律的分支随机游走，研究其范围线性增长率，扩展分支容量至α-稳定情形并估计球的对应容量。