Strong local uniqueness for the vacant set of random interlacements

TL;DR

This work proves sharpness of the percolation transition for the vacant set $\mathcal{V}^u$ of random interlacements on $\mathbb{Z}^d$ ($d\ge 3$) by establishing a strong local uniqueness (SLU) property in the entire supercritical regime $0<u<u_*$. The authors introduce a pseudo-insertion tolerance and a multi-scale excursion-packet renormalization framework, enabling renormalizable control of non-monotone events and a delicate exploration/gluing mechanism via good encounter points. These innovations yield quenched results for the infinite cluster, including a quenched invariance principle for the random walk on $\mathcal{C}_\infty^u$, an isoperimetric-type inequality, and refined bounds on the truncated two-point function $\tau_u^{\mathrm{tr}}(x)$ across non-critical $u$, extending previous near-critical and perturbative analyses to the full supercritical range. The methodology hinges on a coherent combination of restricted insertion tolerance, excursion-based renormalization, and a carefully designed coarse-graining scheme that propagates local properties to macroscopic scales, ultimately establishing the sharp phase transition and broad geometric control of the infinite cluster. The results have significant implications for the geometric and probabilistic understanding of random interlacements and related percolation models in high dimensions.

Abstract

We consider the the vacant set $\mathcal{V}^u$ of random interlacements on $\mathbb{Z}^d$ in dimensions $d \ge 3$. For varying intensity $u > 0$, the connectivity properties of $\mathcal V^u$ undergo a percolation phase transition across a critical parameter $u_* \in (0,\infty)$. In this article, we prove that this phase transition is sharp in the supercritical phase $u < u_*$. This follows from a certain strong local uniqueness property (SLU) introduced in the present work, which we prove $\mathcal{V}^u$ satisfies. In itself, this property furnishes the missing ingredient needed to deduce a number of desirable quenched results characterizing the large-scale geometry of the infinite cluster. Moreover, SLU entails a sought-after local and monotone criterion amenable to renormalization arguments below $u_*$.

Theorems & Definitions (61)

• Theorem 1.1
• Lemma 1.2: Restricted insertion tolerance
• Proposition 1.3
• Theorem 1.4
• Remark 1.5
• Proposition 2.1
• Lemma 2.2: Coupling $Z$ and $\widetilde{Z}$
• Proposition 3.1
• Remark 3.2
• proof