Level-set percolation of the Gaussian free field on regular graphs II: Finite expanders

TL;DR

This paper investigates the phase transition in the level sets of the zero-average Gaussian free field on finite regular graphs, showing a critical threshold where large connected components emerge or vanish, depending on the level.

Contribution

It establishes the existence of a phase transition for level set percolation on finite regular graphs, aligning with the infinite tree case, and characterizes component sizes relative to the critical level.

Findings

Below critical level, only logarithmic-sized components exist.

Above critical level, a linear fraction of vertices are in small components.

The critical level matches the infinite $d$-regular tree case.

Abstract

We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\geq 3$. 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 $h$ exhibits a phase transition at level $h_\star$, which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite $d$-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level $h$ does not contain any connected component of larger than logarithmic size whenever $h>h_\star$, and on the contrary, whenever $h<h_\star$, a linear fraction of the vertices is contained in connected components of the level set above level $h$ having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase $h<h_\star$, as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level $h$. The proofs in this article make use of results from the accompanying paper [AC1].