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

We study the level-set of zero-average Gaussian Free Field on a uniform random d-regular graph above an arbitrary level h∈(−∞,h⋆), where h⋆ is percolation threshold GFF tree Td. prove that w.h.p as number n vertices diverges, has unique giant connected component C1(n) size η(h)n+o(n), η(h) probability root percolates in corresponding This gives positive answer to conjecture [4] for most regular graphs. also second largest Θ(logn). Moreover, we show shares following similarities with supercritical Erdős-Rényi graph. First, diameter and typical distance between are Second, 2-core kernel encompass given proportion vertices. Third, local structure branching process conditioned survive, namely cluster Td (in case, it known be Galton-Watson Poisson distribution offspring).