Disconnection by level sets of the discrete Gaussian free field and entropic repulsion

Abstract

We derive asymptotic upper and lower bounds on the large deviation probability that the level set of the Gaussian free field on $\mathbb{Z} ^d$, $d \geq 3$, below a level $\alpha $ disconnects the discrete blow-up of a compact set $A$ from the boundary of the discrete blow-up of a box that contains $A$, when the level set of the Gaussian free field above $\alpha $ is in a strongly percolative regime. These bounds substantially strengthen the results of [21], where $A$ was a box and the convexity of $A$ played an important role in the proof. We also derive an asymptotic upper bound on the probability that the average of the Gaussian free field well inside the discrete blow-up of $A$ is above a certain level when disconnection occurs. The derivation of the upper bounds uses the solidification estimates for porous interfaces that were derived in the work [15] of A.-S. Sznitman and the author to treat a similar disconnection problem for the vacant set of random interlacements. If certain critical levels for the Gaussian free field coincide, an open question at the moment, the asymptotic upper and lower bounds that we obtain for the disconnection probability match in principal order, and conditioning on disconnection lowers the average of the Gaussian free field well inside the discrete blow-up of $A$, which can be understood as entropic repulsion.