Random walks on torus and random interlacements: Macroscopic coupling and phase transition

TL;DR

This paper develops a new coupling method for the trace of random walks on large tori with random interlacements, revealing a phase transition in the structure of the vacant set that aligns with the critical threshold of interlacements.

Contribution

It introduces a macroscopic coupling technique for random walk traces on tori with random interlacements, enabling analysis of large-scale properties and phase transitions.

Findings

Established a coupling valid up to macroscopic scales.

Identified a sharp phase transition in the diameter of the vacant set.

Confirmed the transition threshold matches the critical value of random interlacements.

Abstract

For $d\ge 3$ we construct a new coupling of the trace left by a random walk on a large $d$-dimensional discrete torus with the random interlacements on $\mathbb Z^d$. This coupling has the advantage of working up to macroscopic subsets of the torus. As an application, we show a sharp phase transition for the diameter of the component of the vacant set on the torus containing a given point. The threshold where this phase transition takes place coincides with the critical value $u_*(d)$ of random interlacements on $\mathbb Z^d$. Our main tool is a variant of the soft-local time coupling technique of [PT12].