Guillaume Conchon-Kerjan (University of Bath)

A Gaussian cousin to the Erdős–Rényi graph

Abstract:

A classical way to study the Gaussian Free Field (GFF) on a graph is to take level-sets, i.e. keep only the vertices above a certain threshold. We establish a sharp phase transition on typical large regular graphs: when the threshold goes below a critical value, a huge connected component emerges in the level-set, containing a positive proportion of the vertices, and sharing many similarities with the giant component of the classical Erdős–Rényi model. The argument relies on a fine understanding of the GFF on regular trees, which yields an infinite-type branching process driven by a compact operator.