Random walk on a discrete Heisenberg group

We use the estimate of paths in Z ² enclosing a null algebraic area to compute correction terms on the random walk on certain discrete Heisenberg groups. We obtain that the probability to return at the origin of the simple random walk on this group is $\frac{1}{4n^{2}}+O(\frac{1}{n^{3}})$ .     

On the disconnection of a discrete cylinder by a random walk

We investigate the large N behavior of the time the simple random walk on the discrete cylinder needs to disconnect the discrete cylinder. We show that when d≥2, this time is roughly of order N ^{2 d} and comparable to the cover time of the slice , but substantially larger than the cover timer of the base by the projection of the walk. Further we show that by the time disconnection occurs, a massive ``clogging' typically takes place in the truncated cylinders of height . These mechanisms are in contrast with what happens when d=1.     

Molecular random walk and a symmetry group of the Bogoliubov equation

We consider the statistics of molecular random walks in fluids using the Bogoliubov equation for the generating functional of the distribution functions. We obtain the symmetry group of this equation and its solutions as functions of the medium density. It induces a series of exact relations between the probability distribution of the total path of a walking test particle and its correlations with the environment and consequently imposes serious constraints on the possible form of the path distribution. In particular, the Gaussian asymptotic form of the distribution is definitely forbidden (even for the Boltzmann-Grad gas), but the diffusive asymptotic form with power-law tails (cut off by the ballistic flight length) is allowed.     

Dimensional properties of the harmonic measure for a random walk on a hyperbolic group

This paper deals with random walks on isometry groups of Gromov hyperbolic spaces, and more precisely with the dimension of the harmonic measure associated with such a random walk. We first establish a link of the form between the dimension of the harmonic measure, the asymptotic entropy of the random walk and its rate of escape . Then we use this inequality to show that the dimension of this measure can be made arbitrarily small and deduce a result on the type of the harmonic measure.

     

Distribution of passage time for a random walk on a finite solvable group

Translated from Matematicheskii Zhurnal, Vol. 41, No. 10, pp. 1395–1398, October, 1989.     

The ratio set of the harmonic measure of a random walk on a hyperbolic group

We consider the harmonic measure on the Gromov boundary of a non-amenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio set by showing that it is generated by certain values of the Martin kernel. In particular, we show that the equivalence relation is never of type III₀.     

Quenched invariance principle for simple random walk on discrete point processes

We consider the simple random walk on random graphs generated by discrete point processes. This random walk moves on graphs whose vertex set is a random subset of a cubic lattice and whose edges are lines between any consecutive vertices on lines parallel to each coordinate axis. Under the assumption that the discrete point processes are finitely dependent and stationary, we prove that the quenched invariance principle holds, i.e., for almost every configuration of the point process, the path distribution of the walk converges weakly to that of a Brownian motion.     

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     

The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus

Let x and y be points chosen uniformly at random from ${\mathbb {Z}_n^4}$ , the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order n ²(log n)^1/6, resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on ${\mathbb {Z}_n^4}$ is the Brownian continuum random tree of Aldous. Our proofs use the techniques developed by Peres and Revelle, who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d-dimensional discrete torus for d ≥ 5, in combination with results of Lawler concerning intersections of four-dimensional random walks.     

Products of random varibles depending on a random walk

LetX=(X _n ) _n0 denote an irreducible random walk (ergodic in the sense of [7]) on a compact metrizable abelian groupG. In this paper we characterize completely the limit distributions of the productsY _n =X ₀...X _n . In particular we find necessary and sufficient conditions forX and/orG to imply that the products are asymptotically equidistributed in the mean, i. e. {im171-1} holds for all open,m _G -regular subsetsA ofG (m _G : normalized Haar measure).—For example ifG is monothetic and connected or ifX is asymptotically equidistributed (not merely in the mean) then the products are asymptotically equidistributed in the mean.Dedicated to Prof. Dr. L. Schmetterer on his 60^th Birthday     

Slow movement of a random walk on the range of a random walk in the presence of an external field

In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions ( $d\ge 5$ ). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon as a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai’s regime. Via this approach, a corresponding aging result is also proved.     

On the average of a random walk

Let {S_n, n ϵ N)} be a simple random walk and denote by A_n its time average: A_n = (S₁+ …+S_n)/n. We give an integral test for the lower bound on A_n, thus giving an affirmative answer to a conjecture of P. Erdös (private communication) that A_n will return to a fixed region around the origin infinitely often with probability 1 in 1 dimension whereas in 2 or more dimensions it will return only finitely many times.     

Once edge-reinforced random walk on a tree

 We consider a nearest neighbor walk on a regular tree, with transition probabilities proportional to weights or conductances of the edges. Initially all edges have weight 1, and the weight of an edge is increased to $c > 1$ when the edge is traversed for the first time. After such a change the weight of an edge stays at $c$ forever. We show that such a walk is transient for all values of $c \ge 1$, and that the walk moves off to infinity at a linear rate. We also prove an invariance principle for the height of the walk. Received: 6 March 2001 / Revised version: 16 July 2001 / Published online: 15 March 2002     

A remark on the CLT for a random walk in a random environment

We give a simplified proof, using elementary methods only, of the almost-sure central limit theorem (CLT) in any dimension for a Markov model of a random walk in a random environment introduced in [BMP].Mathematics Subject Classification (2000): 60F05, 60K37Revised version: 29 January 2004