Transience of the vacant set for near-critical random interlacements in high dimensions

Abstract

The model of random interlacements is a one-parameter family ${\mathcal{I}}^u,$ $u\ge 0,$ of random subsets of ${\mathbb{Z}}^d,$ which locally describes the trace of simple random walk on a $d$-dimensional torus run up to time $u$ times its volume. Its complement, the so-called vacant set ${\mathcal{V}}^u$, has been shown to undergo a non-trivial percolation phase-transition in $u;$ i.e., there exists $u_{\ast }(d)\in (0,\infty )$ such that for $u\in [0,u_{\ast }(d))$ the vacant set ${\mathcal{V}}^u$ contains a unique infinite connected component ${\mathcal{V}}_{\infty }^u,$ while for $u>u_{\ast }(d)$ it consists of finite connected components. Sznitman \cite{SZ11,SZ11B} showed that $u_{\ast }(d)\sim \log d,$ and in this article we show the existence of $u(d)>0$ with $\frac{u(d)}{u_{\ast }(d)}\to 1$ as $d\to \infty$ such that ${\mathcal{V}}_{\infty }^u$ is transient for all $u\in [0,u(d)).$

arXiv comment20 pages