Disconnection and level-set percolation for the Gaussian free field

TL;DR

This paper investigates the percolation properties of the Gaussian free field on high-dimensional integer lattices, providing large deviation estimates for disconnection events and relating these to random interlacements and simple random walks.

Contribution

It introduces new large deviation bounds for the disconnection of large boxes by the Gaussian free field's level sets, connecting percolation, random interlacements, and random walks.

Findings

Derived large deviation estimates for disconnection probabilities.

Established connections between Gaussian free field percolation and random interlacements.

Provided asymptotic upper bounds for disconnection events.

Abstract

We study the level-set percolation of the Gaussian free field on Z^d, d bigger or equal to 3. We consider a level alpha such that the excursion-set of the Gaussian free field above alpha percolates. We derive large deviation estimates on the probability that the excursion-set of the Gaussian free field below the level alpha disconnects a box of large side-length from the boundary of a larger homothetic box. It remains an open question whether our asymptotic upper and lower bounds are matching. With the help of a recent work of Lupu, see arXiv:1402.0298, we are able to infer some asymptotic upper bounds for similar disconnection problems by random interlacements, or by simple random walk.