Phase transition for the vacant set of random walk and random interlacements

TL;DR

This paper proves a phase transition in the structure of the vacant set left by a random walk on a high-dimensional torus, showing a critical point where the set shifts from having a giant component to scattering into tiny parts.

Contribution

It establishes the precise critical value for the phase transition of the vacant set of random walk and connects it to the critical point of random interlacements, confirming long-standing conjectures.

Findings

Existence of a non-trivial phase transition at a critical parameter u_*

Giant component for u < u_* with high probability

Scattering into small components for u > u_*

Abstract

We consider the set of points visited by the random walk on the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, for $d \geq 3$, at times of order $uN^d$, for a parameter $u>0$ in the large-$N$ limit. We prove that the vacant set left by the walk undergoes a phase transition across a non-degenerate critical value $u_* = u_*(d)$, as follows. For all $u< u_*$, the vacant set contains a giant connected component with high probability, which has a non-vanishing asymptotic density and satisfies a certain local uniqueness property. In stark contrast, for all $u> u_*$ the vacant set scatters into tiny connected components. Our results further imply that the threshold $u_*$ precisely equals the critical value, introduced by Sznitman in arXiv:0704.2560, which characterizes the percolation transition of the corresponding local limit, the vacant set of random interlacements on $\mathbb{Z}^d$. Our findings also yield the analogous infinite-volume result, i.e. the long purported equality of three critical parameters $\bar u$, $u_*$ and $u_{**}$ naturally associated to the vacant set of random interlacements.