Anatomy of a Gaussian giant: supercritical level-sets of the free field on regular graphs

Abstract

We study the level-set of the zero-average Gaussian Free Field on a uniform random d-regular graph above an arbitrary level $h\in (-\mathrm{\infty },h_{\star })$, where $h_{\star }$ is the level-set percolation threshold of the GFF on the d-regular tree ${\mathbb{T}}_d$. We prove that w.h.p as the number n of vertices of the graph diverges, the GFF has a unique giant connected component ${\mathcal{C}}_1^{(n)}$ of size $\mathrm{\eta }(h)n+o(n)$, where $\mathrm{\eta }(h)$ is the probability that the root percolates in the corresponding GFF level-set on ${\mathbb{T}}_d$. This gives a positive answer to the conjecture of [4] for most regular graphs. We also prove that the second largest component has size $\mathrm{\Theta }(logn)$.

Moreover, we show that ${\mathcal{C}}_1^{(n)}$ shares the following similarities with the giant component of the supercritical Erdős-Rényi random graph. First, the diameter and the typical distance between vertices are $\mathrm{\Theta }(logn)$. Second, the 2-core and the kernel encompass a given positive proportion of the vertices. Third, the local structure is a branching process conditioned to survive, namely the level-set percolation cluster of the root in ${\mathbb{T}}_d$ (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).