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

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 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 C⁽ⁿ⁾ 1 of size η(h)n + o(n), where η(h) is the probability that the root percolates in the corresponding GFF level-set on 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 Θ(log n). Moreover, we show that C⁽ⁿ⁾ 1 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(log n). 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 T_d (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).