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

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 (Chiarini and Nitzschner (2018)). Our proofs rely crucially on the “solidification estimates” developed in (Nitzschner and Sznitman (2017)).