We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ), d ≥ 3, until uN time steps, u > 0, and the model of random interlacements recently introduced by Sznitman [9]. In particular, we show that for large N , the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to t...

a random walk on a lattice is one of the most fundamental models in probability theory. when the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (rwre). the basic questions such as the law of large numbers (lln), the central limit theorem (clt), and the large deviation principle (ldp) are no...

Benjamini, Lyons and Schramm [Random Walks and Discrete Potential Theory (1999) 56–84] considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. In this paper we solve several problems raised by those authors. The anchored expansion constant is a variant of the Cheeger constant; its positivity implies positive lower speed for the...

G. R. Grimmett and S. N. Winkler Abstra t. We consider three probability measures on subsets of edges of a given finite graph G, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a uniform forest, and a related conjecture is posed for a uniform connec...

We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R2. Specifically, for each > 0 let H be the subgraph of Z2 whose vertices lie in a fixed rectilinear polygon U . Let N (H ) denote the number of vertices of H and B(H ) the number of vertices on the boundary (the outer face). Then the log of the determinant of the Laplacian on H has the f...

Let $ G be a connected semisimple real Lie group with finite center, and \mu probability measure on whose support generates Zariski-dense subgroup of $. We consider the right $-random walk show that each random trajectory spends most its time at bounded distance well-chosen Weyl chamber. infer if has rank one, first moment, then for any discrete \Lambda\subseteq $, $-walk geodesic flow \Lambda ...

We consider discrete-time nearest-neighbor quantum walks on random environments in one dimension. Using the method based on a path counting, we present both quenched and annealed weak limit theorems for the quantum walk.

The strong interaction limit of the discrete-time weakly self-avoiding walk (or Domb–Joyce model) is trivially seen to be the usual strictly self-avoiding walk. For the continuous-time weakly self-avoiding walk, the situation is more delicate, and is clarified in this paper. The strong interaction limit in the continuous-time setting depends on how the fugacity is scaled, and in one extreme lea...

We look at two possible routes to classical behavior for the discrete quantum random walk on the line: decoherence in the quantum “coin” which drives the walk, or the use of higher-dimensional coins to dilute the effects of interference. We use the position variance as an indicator of classical behavior, and find analytical expressions for this in the long-time limit; we see that the multicoin ...