On the fragmentation of a torus by random walk

TL;DR

This paper analyzes how a simple random walk on a high-dimensional torus creates a fragmented vacant set, revealing phase transitions between small and large components, and establishing uniqueness of the large component in dimensions five and above.

Contribution

It proves the existence of two phases in the vacant set and establishes the uniqueness of the macroscopic component in high dimensions, solving open problems in the field.

Findings

For large u, all vacant set components are small, at most a power of log(N).

For small u, a large, non-degenerate component exists occupying a positive volume.

In dimensions d ≥ 5, the large component is unique with high probability.

Abstract

We consider a simple random walk on a discrete torus (Z/NZ)^d with dimension d at least 3 and large side length N. For a fixed constant u > 0, we study the percolative properties of the vacant set, consisting of the set of vertices not visited by the random walk in its first [uN^d] steps. We prove the existence of two distinct phases of the vacant set in the following sense: if u > 0 is chosen large enough, all components of the vacant set contain no more than a power of log(N) vertices with high probability as N tends to infinity. On the other hand, for small u > 0, there exists a macroscopic component of the vacant set occupying a non degenerate fraction of the total volume N^d. In dimensions d at least 5, we additionally prove that this macroscopic component is unique, by showing that all other components have volumes of order at most a power of log(N). Our results thus solve open problems posed by Benjamini and Sznitman in arXiv:math/0610802, who studied the small u regime in high dimension. The proofs are based on a coupling of the random walk with random interlacements on Z^d. Among other techniques, the construction of this coupling employs a refined use of discrete potential theory. By itself, this coupling strengthens a result in arXiv:0802.3654.