Asymptotic entropy and green speed for random walks on countable groups

We study asymptotic properties of the Green metric associated with transient random walks on countable groups. We prove that the rate of escape of the random walk computed in the Green metric equals its asymptotic entropy. The proof relies on integral representations of both quantities with the extended Martin kernel. In the case of finitely generated groups, where this result is known (Benjamini and Peres [Probab. Theory Related Fields 98 (1994) 91--112]), we give an alternative proof relying on a version of the so-called fundamental inequality (relating the rate of escape, the entropy and the logarithmic volume growth) extended to random walks with unbounded support.Comment: Published in at http://dx.doi.org/10.1214/07-AOP356 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A crossover for the bad configurations of random walk in random scenery

In this paper, we consider a random walk and a random color scenery on Z. The increments of the walk and the colors of the scenery are assumed to be i.i.d. and to be independent of each other. We are interested in the random process of colors seen by the walk in the course of time. Bad configurations for this random process are the discontinuity points of the conditional probability distribution for the color seen at time zero given the colors seen at all later times. We focus on the case where the random walk has increments 0, +1 or -1 with probability epsilon, (1-epsilon)p and (1-epsilon)(1-p), respectively, with p in [1/2,1] and epsilon in [0,1), and where the scenery assigns the color black or white to the sites of Z with probability 1/2 each. We show that, remarkably, the set of bad configurations exhibits a crossover: for epsilon=0 and p in (1/2,4/5) all configurations are bad, while for (p,epsilon) in an open neighborhood of (1,0) all configurations are good. In addition, we show that for epsilon=0 and p=1/2 both bad and good configurations exist. We conjecture that for all epsilon in [0,1) the crossover value is unique and equals 4/5. Finally, we suggest an approach to handle the seemingly more difficult case where epsilon>0 and p in [1/2,4/5), which will be pursued in future work.Comment: Published in at http://dx.doi.org/10.1214/11-AOP664 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Harmonic measures versus quasiconformal measures for hyperbolic groups

We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as \qc measures on the boundary of the group.Comment: Besides minor modifications, we provide a new proof that the harmonic measure of a finitely supported random walk on a Fuchsian group with cusps is singular. 52 p

The dispersion time of random walks on finite graphs

We study two random processes on an $n$-vertex graph inspired by the internal diffusion limited aggregation (IDLA) model. In both processes $n$ particles start from an arbitrary but fixed origin. Each particle performs a simple random walk until first encountering an unoccupied vertex, and at which point the vertex becomes occupied and the random walk terminates. In one of the processes, called $\textit{Sequential-IDLA}$, only one particle moves until settling and only then does the next particle start whereas in the second process, called $\textit{Parallel-IDLA}$, all unsettled particles move simultaneously. Our main goal is to analyze the so-called dispersion time of these processes, which is the maximum number of steps performed by any of the $n$ particles. In order to compare the two processes, we develop a coupling that shows the dispersion time of the Parallel-IDLA stochastically dominates that of the Sequential-IDLA; however, the total number of steps performed by all particles has the same distribution in both processes. This coupling also gives us that dispersion time of Parallel-IDLA is bounded in expectation by dispersion time of the Sequential-IDLA up to a multiplicative $\log n$ factor. Moreover, we derive asymptotic upper and lower bound on the dispersion time for several graph classes, such as cliques, cycles, binary trees, $d$-dimensional grids, hypercubes and expanders. Most of our bounds are tight up to a multiplicative constant.ERC Grant Dynamic Marc