Soft local times and decoupling of random interlacements

TL;DR

This paper introduces a decoupling technique for the random interlacement process in high-dimensional lattices, enabling analysis of connectivity properties and long path probabilities by controlling interactions between disjoint sets.

Contribution

It develops a novel decoupling method using soft local times and introduces a way to smoothen discrete sets for regular equilibrium measures, advancing the understanding of random interlacements.

Findings

Decoupling holds for sets with small mutual distance relative to their size.

Probability of long paths avoiding interlacements is exponentially small above a critical threshold.

New auxiliary results on comparing Markov chain traces and smoothing sets are independently valuable.

Abstract

In this paper we establish a decoupling feature of the random interlacement process I^u in Z^d, at level u, for d \geq 3. Roughly speaking, we show that observations of I^u restricted to two disjoint subsets A_1 and A_2 of Z^d are approximately independent, once we add a sprinkling to the process I^u by slightly increasing the parameter u. Our results differ from previous ones in that we allow the mutual distance between the sets A_1 and A_2 to be much smaller than their diameters. We then provide an important application of this decoupling for which such flexibility is crucial. More precisely, we prove that, above a certain critical threshold u**, the probability of having long paths that avoid I^u is exponentially small, with logarithmic corrections for d=3. To obtain the above decoupling, we first develop a general method for comparing the trace left by two Markov chains on the same state space. This method is based in what we call the soft local time of a chain. In another crucial step towards our main result, we also prove that any discrete set can be "smoothened" into a slightly enlarged discrete set, for which its equilibrium measure behaves in a regular way. Both these auxiliary results are interesting in themselves and are presented independently from the rest of the paper.