Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

TL;DR

This paper studies the behavior of the Gaussian free field conditioned on disconnection events caused by level-sets, revealing an entropic push-down and a macroscopic pinning effect influenced by harmonic potentials.

Contribution

It provides large deviation bounds and a profile description for the Gaussian free field under disconnection conditioning, extending understanding of entropic repulsion phenomena.

Findings

Disconnection causes an entropic push-down proportional to harmonic potential.

The field is macroscopically pinned but locally retains Gaussian free field structure.

Disconnection influences the entire lattice due to slow decay of correlations.

Abstract

We investigate level-set percolation of the discrete Gaussian free field on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the level-set of the Gaussian free field below a level $\alpha$ disconnects the discrete blow-up of a compact set $A$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the Gaussian free field deviate from a specific multiple of the harmonic potential of $A$, when disconnection occurs. These bounds, combined with the findings of the recent article [12], show that conditionally on disconnection, the Gaussian free field experiences an entropic push-down proportional to the harmonic potential of $A$. In particular, due to the slow decay of correlations, the disconnection event affects the field on the whole lattice. Furthermore, we provide a certain 'profile' description for the field in the presence of disconnection. We show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of $A$, it locally retains the structure of a Gaussian free field shifted by a constant value. Our proofs rely crucially on the 'solidification estimates' developed in arXiv:1706.07229 by A.-S. Sznitman and the second author.