## Abstract

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 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 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 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 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 T_{d} (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).

Original language | English |
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Article number | 35 |

Pages (from-to) | 1-60 |

Journal | Electronic Journal of Probability |

Volume | 28 |

Early online date | 27 Feb 2023 |

DOIs | |

Publication status | Published - 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: galck20@bath.ac.uk

## Keywords

- Gaussian free field
- percolation
- random graphs

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty