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

Guillaume Conchon-Kerjan

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We study the level-set of the zero-average Gaussian Free Field on a uniform random d-regular graph above an arbitrary level h ∈ (−∞, h), where h is the level-set percolation threshold of the GFF on the d-regular tree Td. We prove that w.h.p as the number n of vertices of the graph diverges, the GFF has a unique giant connected component C(n) 1 of size η(h)n + o(n), where η(h) is the probability that the root percolates in the corresponding GFF level-set on Td. This gives a positive answer to the conjecture of [4] for most regular graphs. We also prove that the second largest component has size Θ(log n). Moreover, we show that C(n) 1 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 Θ(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 Td (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).

Original languageEnglish
Article number35
Pages (from-to)1-60
JournalElectronic Journal of Probability
Early online date27 Feb 2023
Publication statusPublished - 31 Dec 2023

Bibliographical note

Funding Information:
*GCK is grateful to EPSRC for support through the grant EP/V00929X/1. †University of Bath, United Kingdom. E-mail:


  • Gaussian free field
  • percolation
  • random graphs

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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