Simple transient random walks in one-dimensional random environment: the central limit theorem

We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the central limit theorem (CLT) holds for the position of the walk. Verifying these conditions leads to a complete solution of the problem in the case of independent identically distributed environments as well as in the case of uniformly ergodic (and thus also weakly mixing) environments.      

Persistent random walks in random environment

Summary Weak convergence of a class of functionals of PRWRE is proved. As a consequence CLT is obtained for the normed trajectory.Work supported by the Central Research Fund of the Hungarian Academy of Sciences (Grant No. 476/82).     

Localization for branching random walks in random environment

We consider branching random walks in d$d$-dimensional integer lattice with time–space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If d≥3$d\ge 3$ and the environment is “not too random”, then, the total population grows as fast as its expectation with strictly positive probability. If, on the other hand, d≤2$d\le 2$, or the environment is “random enough”, then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of “replica overlap”. We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.     

On the range of reversible random walks onZ 2 in a random environment

In this paper, a weak law of large numbers is obtained for the range of two dimensional reversible random walk in a random environment.Partly supported by NSF of China.     

Non parametric estimation for random walks in random environment

We investigate the problem of estimating the cumulative distribution function (c.d.f.) $F$ of a distribution $\nu$ from the observation of one trajectory of the random walk in i.i.d. random environment with distribution $\nu$ on $\mathbb{Z}$. We first estimate the moments of $\nu$, then combine these moment estimators to obtain a collection of estimators ${\left({\stackrel{?}{F}}_n^M\right)}_{M\ge 1}$ of $F$, our final estimator is chosen among this collection by Goldenshluger–Lepski’s method. This estimator is easily computable. We derive convergence rates for this estimator depending on the Hölder regularity of $F$ and on the divergence rate of the walk. Our rate is minimal when the chain realizes a trade-off between a fast exploration of the sites, allowing to get more information and a larger number of visits of each site, allowing a better recovery of the environment itself.     

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.     

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.     

Quenched invariance principle for random walks in balanced random environment

We consider random walks in a balanced random environment in ${\mathbb{Z}^d}$ , d?≥ 2. We first prove an invariance principle (for d?≥ 2) and the transience of the random walks when d?≥ 3 (recurrence when d?=?2) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments.     

一类时间随机环境中随机游动

利用概率母函数方法,通过对一类时间随机环境中随机游动首中时性质的研究,得到了该随机游动的常返准则和一个强大数定律.     

Ellipticity criteria for ballistic behavior of random walks in random environment

We introduce ellipticity criteria for random walks in i.i.d. random environments under which we can extend the ballisticity conditions of Sznitman and the polynomial effective criteria of Berger, Drewitz and Ramírez originally defined for uniformly elliptic random walks. We prove under them the equivalence of Sznitman’s \((T')\) condition with the polynomial effective criterion \((P)_M\) , for \(M\) large enough. We furthermore give ellipticity criteria under which a random walk satisfying the polynomial effective criterion, is ballistic, satisfies the annealed central limit theorem or the quenched central limit theorem.     

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.     

Slowdown estimates and central limit theorem for random walks in random environment

This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in random environment on ℤ ^d , when d≥2. We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed in [25] plays an important role in the investigation of both problems. This article also improves the previous work of the author [24], concerning estimates of probabilities of slowdowns for walks which are neutral or biased to the right. Received May 31, 1999 / final version received January 18, 2000?Published online April 19, 2000     

Large deviations for random walks in a mixing random environment and other (non‐Markov) random walks

We extend a recent work by S. R. S. Varadhan [8] on large deviations for random walks in a product random environment to include more general random walks on the lattice. In particular, some reinforced random walks and several classes of random walks in Gibbs fields are included. © 2004 Wiley Periodicals, Inc.     

An effective criterion for ballistic behavior of random walks in random environment

 We investigate multi-dimensional random walks in random environment. Specifically, we provide an effective criterion which can be checked by inspection of the environment in a finite box, and implies a ballistic strong law of large numbers, a central limit theorem and several large deviation controls. With the help of this criterion, we provide an example of a ballistic multi-dimensional walk which does not fulfill the criterion introduced by Kalikow in [4]. The present work complements the results of [9], where a certain condition (T) was introduced, and confirms the interest of this condition. Received: 5 October 2000/ Revised version: 24 May 2001 Published online: 22 February 2002     

Quasimorphisms,random walks,and transient subsets in countable groups

We study the interrelations between the theory of quasimorphisms and the theory of random walks on groups, and establish the following transience criterion for subsets of groups: if a subset of a countable group has bounded images under any three linearly independent homogeneous quasimorphisms on the group, the this subset is transient for all nondegenerate random walks on the group. From this it follows, by results of M. Bestvina, K. Fujiwara, J. Birman, W. Menasco, and others, that, in a certain sense, generic elements in the mapping class groups of surfaces are pseudo-Anosov, generic braids in Artin’s braid groups represent prime links and knots, generic elements in the commutant of every nonelementary hyperbolic group have large stable commutator length, etc. Bibliography: 20 titles.     

Symmetric exclusion on random sets and a related problem for random walks in random environment

Summary We study symmetric exclusion on a random set, where the underlying kernelp(x, y) is strictly positive. The random set is generated by Bernoulli experiments with success probabilityq.We prove that for certain values of the involved parameters the transport of particles through the system is drastically different from the classical situation on . In dimension one and the transport of particles occurs on a nonclassical scale and is (on a macroscopic scale)not governed by the heat equation as in the case:r<|log(1-q)| on a random set, or in the classical situation on .The reason for this behaviour is, that a random walk with jump ratesp(x, y) restricted to the random set, converges to Brownian motion in the usual scaling ifr<|log(1-q)| but yields nontrivial limit behaviour only in the scalingxu _-1 x,tu _1+a t(>) if + >r > |log(1-q)|. We calculate and study the limiting processes for the various scalings for fixed random sets. We shortly discuss the caser=+, here in general a great variety of scales yields nontrivial limits.Finally we discuss the case of a stationary random set.