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Anatomy of a Gaussian giant: supercritical level-sets of the free field on regular graphs. (English) Zbl 1533.60170

Summary: We study the level-set of the zero-average Gaussian Free Field on a uniform random \(d\)-regular graph above an arbitrary level \(h\in (-\infty,h_{\star})\), where \(h_{\star}\) is the level-set percolation threshold of the GFF on the \(d\)-regular tree \(\mathbb{T}_d \). We prove that w.h.p as the number \(n\) of vertices of the graph diverges, the GFF has a unique giant connected component \(\mathcal{C}_1^{(n)}\) of size \(\eta (h)n+o(n)\), where \(\eta (h)\) is the probability that the root percolates in the corresponding GFF level-set on \(\mathbb{T}_d\). This gives a positive answer to the conjecture of [4] for most regular graphs. We also prove that the second largest component has size \(\Theta (\log n)\).
Moreover, we show that \({\mathcal{C}_1^{(n)}}\) 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 \(\Theta (\log n)\). 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 \({\mathbb{T}_d} \) (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G15 Gaussian processes
60C05 Combinatorial probability
05C80 Random graphs (graph-theoretic aspects)
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