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∈(−∞,h⋆), where h⋆ is the level-set percolation threshold of the GFF on the d-regular tree Td. We prove that w.h.p as the number n of vertices of the graph diverges, the GFF has a unique giant connected component C1(n) of size η(h)n+o(n), where η(h) is the probability that the root percolates in the corresponding GFF level-set on Td. This gives a positive answer to the conjecture of [4] for most regular graphs. We also prove that the second largest component has size Θ(logn). Moreover, we show that C1(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 Θ(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 Td (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).