Isomorphism of random walks

We show that any two aperiodic, recurrent random walks on the integers whose jump distributions have finite seventh moment, are isomorphic as infinite measure preserving transformations. The method of proof involved uses a notion of equivalence of renewal sequences, and the “relative” isomorphism of Bernoulli shifts respecting a common state lumping with the same conditional entropy. We also prove an analogous result for random walks on the two dimensional integer lattice.     

Multi-excited random walks on integers

We introduce a class of nearest-neighbor integer random walks in random and non-random media, which includes excited random walks considered in the literature. At each site the random walker has a drift to the right, the strength of which depends on the environment at that site and on how often the walker has visited that site before. We give exact criteria for recurrence and transience and consider the speed of the walk.Most of this work was done while the author was Szegö Assistant Professor at Stanford University.     

Maxima of branching random walks vs. independent random walks

In recent years several authors have obtained limit theorems for the location of the right most particle in a supercritical branching random walk. In this paper we will consider analogous problems for an exponentially growing number of independent random walks. A comparison of our results with the known results of branching random walk then identifies the limit behaviors which are due to the number of particles and those which are determined by the branching structure.     

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.     

Branching random walks II

A general method is developed with which various theorems on the mean square convergence of functionals of branching random walks are proven. The results cover extensions and generalizations of classical central limit analogues as well as a result of a different type.     

Branching random walks I

A general method is developed with which various theorems on the mean square convergence of functionals of branching random walks are proven. The results cover extensions and generalizations of classical central limit analogues as well as a result of a different type.     

Quantum random walks and their convergence to Evans-Hudson flows

Using coordinate-free basic operators on toy Fock spaces, quantum random walks are defined following the ideas of Attal and Pautrat. Extending the result for one dimensional noise, strong convergence of quantum random walks associated with bounded structure maps to Evans-Hudson flow is proved under suitable assumptions. Starting from the bounded generator of a given uniformly continuous quantum dynamical semigroup on a von Neumann algebra, we have constructed quantum random walks which converges strongly and the strong limit gives an Evans-Hudson dilation for the semigroup.     

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.     

On random walks for Pollard's rho method

We consider Pollard's rho method for discrete logarithm computation. Usually, in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated. We show that this assumption does not hold for the walk originally suggested by Pollard: its performance is worse than in the random case. We study alternative walks that can be efficiently applied to compute discrete logarithms. We introduce a class of walks that lead to the same performance as expected in the random case. We show that this holds for arbitrarily large prime group orders, thus making Pollard's rho method for prime group orders about faster than before.

     

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     

Snakes and perturbed random walks

We study some properties of random walks perturbed at extrema, which are generalizations of the walks considered, e.g., by Davis (1999) and Tóth (1996). This process can also be viewed as a version of an excited random walk, recently studied by many authors. We obtain several properties related to the range of the process with infinite memory and prove the strong law, the central limit theorem, and the criterion for the recurrence of the perturbed walk with finite memory. We also state some open problems. Our methods are predominantly combinatorial and do not involve complicated analytic techniques.     

Stratified random walks on the n-cube

In this paper we present a method for analyzing a general class of random walks on the n-cube (and certain subgraphs of it). These walks all have the property that the transition probabilities depend only on the level of the point at which the walk is. For these walks, we derive sharp bounds on their mixing rates, i.e., the number of steps required to guarantee that the resulting distribution is close to the (uniform) stationary distribution. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 199–222, 1997     

Scaling exponents of random walks in random sceneries

We compute the exact asymptotic normalizations of random walks in random sceneries, for various null recurrent random walks to the nearest neighbours, and for i.i.d., centered and square integrable random sceneries. In each case, the standard deviation grows like n with . Here, the value of the exponent is determined by the sole geometry of the underlying graph, as opposed to previous examples, where this value reflected mainly the integrability properties of the steps of the walk, or of the scenery. For discrete Bessel processes of dimension d[0;2[, the exponent is . For the simple walk on some specific graphs, whose volume grows like n^d for d[1;2[, the exponent is =1−d/4. We build a null recurrent walk, for which without logarithmic correction. Last, for the simple walk on a critical Galton–Watson tree, conditioned by its nonextinction, the annealed exponent is . In that setting and when the scenery is i.i.d. by levels, the same result holds with .     

Collision problems of random walks in two-dimensional time

The collision problems of two-parameter random walks are studied. That is, some criteria have been established in terms of the characteristic functions of two or more mutually independent random walks in order to determine if they meet infinitly often in certain restricted time sets.     

On mixing of certain random walks, cutoff phenomenon and sharp threshold of random matroid processes

In this paper we define and analyze convergence of the geometric random walks, which are certain random walks on vector spaces over finite fields. We show that the behavior of such walks is given by certain random matroid processes. In particular, the mixing time is given by the expected stopping time, and the cutoff is equivalent to sharp threshold. We also discuss some random geometric random walks as well as some examples and symmetric cases.     

Limit theorems for reinforced random walks on certain trees

Consider a linearly edge-reinforced random walk defined on the b-ary tree, b≥70. We prove the strong law of large numbers for the distance of this process from the root. We give a sufficient condition for this strong law to hold for general edge-reinforced random walks and random walks in a random environment. We also provide a central limit theorem. Supported in part by a Purdue Research Foundation fellowship this work is part of the author's PhD thesis.     

Local approximations of Markov random walks by diffusions

We consider triangular arrays of Markov random walks that can be approximated by an accompanying sequence of diffusion processes. We give uniform bounds for approximation of scaled transition probabilities by transition densities of the diffusion process. In particular, we state local limit theorems for the case that the Markov random walks converge weakly to a diffusion process.     

Commute times of random walks on trees

In this paper we provide exact formula for the commute times of random walks on spherically symmetric random trees. Using this formula we sharpen some of the results presented in Al-Awadhi et al. to the form of equalities rather than inequalities.     

Asymptotic direction for random walks in random environments

In this paper we study the existence of an asymptotic direction for random walks in random i.i.d. environments (RWRE). We prove that if the set of directions where the walk is transient contains a non-empty open set, the walk admits an asymptotic direction. The main tool to obtain this result is the construction of a renewal structure with cones. We also prove that RWRE admits at most two opposite asymptotic directions.     

Clairvoyant scheduling of random walks

Two infinite walks on the same finite graph are called compatible if it is possible to introduce delays into them in such a way that they never collide. Years ago, Peter Winkler asked the question: for which graphs are two independent random walks compatible with positive probability. Up to now, no such graphs were found. We show in this paper that large complete graphs have this property. The question is equivalent to a certain dependent percolation with a power‐law behavior: the probability that the origin is blocked at distance n but not closer decreases only polynomially fast and not, as usual, exponentially. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011