We study the level-set of the zero-average Gaussian Free Field on a uniform random d-regular graph above an arbitrary level , where is the level-set percolation threshold of the GFF on the d-regular tree . We prove that w.h.p as the number n of vertices of the graph diverges, the GFF has a unique giant connected component of size , where is the probability that the root percolates in the corresponding GFF level-set on . This gives a positive answer to the conjecture of  for most regular graphs. We also prove that the second largest component has size .
Moreover, we show that 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 . 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 (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).
GCK is grateful to EPSRC for support through the grant EP/V00929X/1.
Guillaume Conchon-Kerjan. "Anatomy of a Gaussian giant: supercritical level-sets of the free field on regular graphs." Electron. J. Probab. 28 1 - 60, 2023. https://doi.org/10.1214/23-EJP920