Finite range interlacements and couplings

Abstract

In this article, we consider the interlacement set $\mathcal{I}^u$ at level $u>0$ on $\mathbb{Z}^d$, $d \geq3$, and its finite range version $\mathcal{I}^{u,L}$ for $L >0$, given by the union of the ranges of a Poisson cloud of random walks on $\mathbb{Z}^d$ having intensity $u/L$ and killed after $L$ steps. As $L\to \infty$, the random set $\mathcal{I}^{u,L}$ has a non-trivial (local) limit, which is precisely $\mathcal{I}^u$. A natural question is to understand how the sets $\mathcal{I}^{u,L}$ and $\mathcal{I}^{{u}}$ can be related, if at all, in such a way that their intersections with a box of large radius $R$ almost coincide. We address this question, which depends sensitively on $R$, by developing couplings allowing for a similar comparison to hold with very high probability for $\mathcal{I}^{u,L}$ and $\mathcal{I}^{{u'},2L}$, with $u' \approx u$. In particular, for the vacant set $\mathcal{V}^u=\mathbb{Z}^d \setminus \mathcal{I}^u$ with values of $u$ near the critical threshold, our couplings remain effective at scales $R \gg \sqrt{L}$, which corresponds to a natural barrier across which the walks of length $L$ comprised in $\mathcal{I}^{u,L}$ de-solidify inside $B_R$, i.e. lose their intrinsic long-range structure to become increasingly "dust-like". These mechanisms are complementary to the solidification effects recently exhibited in arXiv:1706.07229. By iterating the resulting couplings over dyadic scales $L$, the models $\mathcal{I}^{u,L}$ are seen to constitute a stationary finite range approximation of $\mathcal{I}^u$ at large spatial scales near the critical point $u_*$. Among others, these couplings are important ingredients for the characterization of the phase transition for percolation of the vacant sets of random walk and random interlacements in two upcoming companion articles.