On the range of a random walk in a torus and random interlacements
Abstract
Let a simple random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random subgraph induced by the set of vertices visited by the walk. Distance and mixing bounds for the typical range are proven that are a -iterated log factor from those on the full torus for arbitrary . The proof uses hierarchical renormalization and techniques that can possibly be applied to other random processes in the Euclidean lattice. We use the same technique to bound the heat kernel of a random walk on random interlacements.
Cite
@article{arxiv.1007.1401,
title = {On the range of a random walk in a torus and random interlacements},
author = {Eviatar B. Procaccia and Eric Shellef},
journal= {arXiv preprint arXiv:1007.1401},
year = {2014}
}
Comments
Published in at http://dx.doi.org/10.1214/14-AOP924 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)