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

Abstract

We investigate level-set percolation of the discrete Gaussian free field on ${\mathbb{Z}}^d$, $d\ge 3$, 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.

Accepted in Probability Theory and Related Fields

References in corpus (4)

• The sign clusters of the massless Gaussian free field percolate on ${\mathbb{Z}}^d$, $d⩾3$ (and more)
• On macroscopic holes in some supercritical strongly dependent percolation models
• Geometry of Gaussian free field sign clusters and random interlacements
• Disconnection by level sets of the discrete Gaussian free field and entropic repulsion