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.