On large deviations and intersection of random interlacements

TL;DR

This paper studies large deviations in random interlacements on high-dimensional lattices, focusing on the capacity of the interlacement set and intersection probabilities, revealing entropic repulsion phenomena.

Contribution

It derives large deviation rates for the capacity of interlacement sets and their intersections, and demonstrates entropic repulsion in this context.

Findings

Large deviation rate for small capacity of interlacement set.

Large deviation rate for empty intersection probability.

Conditioned interlacement becomes sparse, illustrating entropic repulsion.

Abstract

We investigate random interlacements on $\mathbb{Z}^d$ with $d \geq 3$, and derive the large deviation rate for the probability that the capacity of the interlacement set in a macroscopic box is much smaller than that of the box. As an application, we obtain the large deviation rate for the probability that two independent interlacements have empty intersections in a macroscopic box. We also prove that conditioning on this event, one of them will be sparse in the box in terms of capacity. This result is an example of the entropic repulsion phenomenon for random interlacements.