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

Abstract

For $d\ge3$, 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 $u_{\star}(d)$ of random interlacements on $\mathbb{Z}^{d}$. Our main tool is a variant of the soft-local time coupling technique of Popov and Teixeira [J. Eur. Math. Soc. (JEMS) 17 (2015) 2545–2593].