On the range of a random walk in a torus and random interlacements

TL;DR

This paper studies the geometric properties of the range of a simple random walk in high-dimensional tori and applies hierarchical renormalization techniques to derive bounds, also extending to random interlacements.

Contribution

It introduces hierarchical renormalization methods to analyze the range of random walks and bounds heat kernels on random interlacements, advancing understanding of these processes.

Findings

Distance and mixing bounds close to those on the full torus

Hierarchical renormalization techniques applicable to other processes

Bounds on heat kernel for 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 $k$-iterated log factor from those on the full torus for arbitrary $k$. 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.