# Entropic repulsion for the occupation-time field of random interlacements conditioned on disconnection

## TL;DR

This paper studies the behavior of the vacant set of random interlacements on high-dimensional lattices, deriving large deviation bounds and revealing an entropic repulsion phenomenon when the set disconnects a compact region.

## Contribution

It provides new large deviation estimates and demonstrates entropic repulsion effects for the occupation-time field conditioned on disconnection, extending previous results to random interlacements.

## Key findings

- Asymptotic large deviation upper bounds for occupation times
- Identification of entropic push in occupation-time profiles
- Extension of entropic repulsion phenomena to random interlacements

## Abstract

We investigate percolation of the vacant set of random interlacements on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the interlacement set at level $u$ disconnects the discrete blow-up of a compact set $A\subseteq \mathbb{R}^d$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the occupation times deviate from a specific function depending on the harmonic potential of $A$, when disconnection occurs. If certain critical levels coincide, which is plausible but open at the moment, these bounds imply that conditionally on disconnection, the occupation-time profile undergoes an entropic push governed by a specific function depending on $A$. Similar entropic repulsion phenomena conditioned on disconnection by level-sets of the discrete Gaussian free field on $\mathbb{Z}^d$, $d \geq 3$, have been obtained by the authors in arxiv:1808.09947. Our proofs rely crucially on the `solidification estimates' developed in arXiv:1706.07229 by A.-S. Sznitman and the second author.