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

TL;DR

This paper establishes asymptotic bounds on the probability that level sets of the Gaussian free field cause disconnection in high-dimensional lattices, extending previous results and exploring entropic repulsion effects.

Contribution

It provides new asymptotic bounds for disconnection probabilities in the Gaussian free field, removing convexity constraints and linking to entropic repulsion phenomena.

Findings

Bounds match in principal order under certain conditions

Disconnection probability decreases exponentially with scale

Conditioning on disconnection influences field averages

Abstract

We derive asymptotic upper and lower bounds on the large deviation probability that the level set of the Gaussian free field on $Z^d$, d bigger or equal to three, below a given level 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 this level is in a strongly percolative regime. These bounds substantially strengthen the results of arXiv:1412.3960, 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 arXiv:1706.07229 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.