Velocity and Lyapounov exponents of some random walks in random environment

We express the asymptotic velocity of random walks in random environment satisfying Kalikow's condition in terms of the Lyapounov exponents which have previously been used in the context of large deviations.     

Self-intersections of random walks on lattices

Let {X _n ^d }_n≥0be a uniform symmetric random walk on Z^d, and Π^(d) (a,b)={X _n ^d ∈ Z^d : a ≤ n ≤ b}. Suppose f(n) is an integer-valued function on n and increases to infinity as n↑∞, and let

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 Z^d${\mathbb{Z}}^d$, d≥1$d\ge 1$. The red particles jump at rate 1 and are in a Poisson equilibrium with density μ$\mu$. The green particle also jumps at rate 1, but uses different transition kernels p^′$p^{\prime }$ and p^″$p^{″}$ depending on whether it sees a red particle or not. It is shown that, in the limit as μ→∞$\mu \to \infty$, the speed of the green particle tends to the average jump under p^′$p^{\prime }$. 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.     

On the speed of a cookie random walk

We consider the model of the one-dimensional cookie random walk when the initial cookie distribution is spatially uniform and the number of cookies per site is finite. We give a criterion to decide whether the limiting speed of the walk is non-zero. In particular, we show that a positive speed may be obtained for just three cookies per site. We also prove a result on the continuity of the speed with respect to the initial cookie distribution.      

Expected hitting and cover times of random walks on some special graphs

We give general bounds (and in some cases exact values) for the expected hitting and cover times of the simple random walk on some special undirected connected graphs using symmetry and properties of electrical networks. In particular we give easy proofs for an N–1H_N-1 lower bound and an N² upper bound for the cover time of symmetric graphs and for the fact that the cover time of the unit cube is Φ(NlogN). We giver a counterexample to a conjecture of Freidland about a general bound for hitting times. Using the electric approach, we provide some genral upper and lower bounds for the expected cover times in terms of the diameter of the graph. These bounds are tight in many instances, particularly when the graph is a tree. © 1994 John Wiley & Sons, Inc.     

On the laws of large numbers for nonnegative random variables

Strong laws of large numbers concerning nonnegative random variables are obtained and then they are utilized to establish stability results, among other things, for sums of pairwise independent random variables and the range of random walks.     

On the range of simple symmetric random walks on the line

This paper is aimed at a detailed study of the behaviors of random walks which is defined by the dyadic expansions of points. More precisely, let $x=({ϵ}_1\left(x\right),{ϵ}_2\left(x\right),\dots )$ be the dyadic expansion for a point $x\in [0,1)$ and $S_n\left(x\right)={\sum }_{k=1}^n(2{ϵ}_k\left(x\right)-1)$, which can be regarded as a simple symmetric random walk on $\mathbb{Z}.$ Denote by $R_n\left(x\right)$ the cardinality of the set $\{S_1\left(x\right),\dots ,S_n\left(x\right)\},$ which is just the distinct position of $x$ passed after $n$ times. The set of points whose behavior satisfies $R_n\left(x\right)\sim c{n}^{\gamma }$ is studied ($c>0$ and $0<\gamma \le 1$ being fixed) and its Hausdorff dimension is calculated.     

On the structure of stable random walks

We show that the Cauchy random walk on the line, and the Gaussian random walk on the plane are similar as infinite measure preserving transformations.     

Asymptotic behavior for random walk in random environment with holding times

In this article, we mainly discuss the asymptotic behavior for multi-dimensional continuous-time random walk in random environment with holding times. By constructing a renewal structure and using the point “environment viewed from the particle”, under General Kalikow's Condition, we show the law of large numbers (LLN) and central limit theorem (CLT) for the escape speed of random walk.     

Random walks in a strongly sparse random environment

The integer points (sites) of the real line are marked by the positions of a standard random walk with positive integer jumps. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity and assuming additionally that the distribution tail of the jumps is regularly varying at infinity we consider a nearest neighbor random walk on the set of integers having jumps $\pm 1$ with probability $1∕2$ at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2019) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.     

Branching random walks with random environments in time

We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the convergence of the free energy n-llog Zn（t）, large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn（t）/E[Zn（t）ξ].     

An intersection property of the simple random walks inZ d

Let {S _d (n)} _n0 be the simple random walk inZ ^d , and ^(d)(a,b)={S _d (n)Z ^d :anb}. Supposef(n) is an integer-valued function and increases to infinity asn tends to infinity, andE _n ^(d) ={^(d)(0,n)^(d)(n+f(n),)}. In this paper, a necessary and sufficient condition to ensureP(E _n ^d) ,i.o.)=0, or 1 is derived ford=3, 4. This problem was first studied by P. Erdös and S.J. Taylor.This work is partly supported by the National Natural Sciences Foundation of China.     

Diffusivity of a random walk on random walks

We consider a random walk with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor with respect to the case of the classical simple random walk without constraint. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 267–283, 2015     

Moments and distribution of the local time of a two-dimensional random walk

Let ?(n,x)$?(n,x)$ be the local time of a random walk on Z²${\mathbb{Z}}^2$. We prove a strong law of large numbers for the quantity _Ln(α)=_∑x∈Z2?(n,x)^α$L_n\left(\alpha \right)={\sum }_{x\in {\mathbb{Z}}^2}?{(n,x)}^{\alpha }$ for all α≥0$\alpha \ge 0$. We use this result to describe the distribution of the local time of a typical point in the range of the random walk.     

Laws of iterated logarithm for transient random walks in random environments

We consider laws of iterated logarithm for one-dimensional transient random walks in random environments. A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environments, including independent identically distributed environments and uniformly ergodic environments.     

Central limit theorems for a branching random walk with a random environment in time

We consider a branching random walk with a random environment in time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The environment is supposed to be independent and identically distributed. For A ?, let Z_n(A) be the number of particles of generation n located in A. We show central limit theorems for the counting measure Z_n(·) with appropriate normalization.     

A mixed random walk on nonnegative matrices: A law of large numbers

In this paper a mixed random walk on nonnegative matrices has been studied. Under reasonable conditions, existence of a unique invariant probability measure and a law of large numbers have been established for such walks.     

Large deviations for hitting times of a random walk in random environment on a strip

We consider a random walk in random environment on a strip, which is transient to the right. The random environment is stationary and ergodic. By the constructed enlarged random environment which was first introduced by Goldsheid （2008）, we obtain the large deviations conditioned on the environment （in the quenched case） for the hitting times of the random walk.