On the Range of a Random Walk In a Torus and Random Interlacements

Details On the Range of a Random Walk In a Torus and Random Interlacements On the Range of a Random Walk In a Torus and Random Interlacements

2010-07-08

arxiv.org/abs/1007.1401v3

Mathematics

Probability

40 pages

Scientific paper

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 the trace of random interlacements.

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