Giant component for the supercritical level-set percolation of the Gaussian free field on regular expander graphs

TL;DR

This paper proves that in large regular expander graphs, the level set of the zero-average Gaussian free field above a certain threshold contains a unique giant component with high probability, extending previous mesoscopic results.

Contribution

It establishes the existence and uniqueness of a giant component in the supercritical phase for the Gaussian free field on regular expander graphs, improving prior mesoscopic findings.

Findings

Giant component exists for all levels below critical threshold h*

The giant component is unique in the supercritical phase

Probability of giant component tends to one as graph size increases

Abstract

We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\ge 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ has a giant component in the whole supercritical phase, that is for all $h<h_\star$, with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [AC20b], where it was shown that a linear fraction of vertices is in mesoscopic components if $h<h_\star$.