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2020 Level-set percolation of the Gaussian free field on regular graphs II: finite expanders
Angelo Abächerli, Jiří Černý
Electron. J. Probab. 25: 1-39 (2020). DOI: 10.1214/20-EJP532

Abstract

We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\geq 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ exhibits a phase transition at level $h_{\star }$, which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite $d$-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level $h$ does not contain any connected component of larger than logarithmic size whenever $h>h_{\star }$, and on the contrary, whenever $h<h_{\star }$, a linear fraction of the vertices is contained in connected components of the level set above level $h$ having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase $h<h_{\star }$, as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level $h$. The proofs in this article make use of results from the accompanying paper [2].

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Angelo Abächerli. Jiří Černý. "Level-set percolation of the Gaussian free field on regular graphs II: finite expanders." Electron. J. Probab. 25 1 - 39, 2020. https://doi.org/10.1214/20-EJP532

Information

Received: 23 November 2019; Accepted: 3 October 2020; Published: 2020
First available in Project Euclid: 24 October 2020

MathSciNet: MR4169171
Digital Object Identifier: 10.1214/20-EJP532

Subjects:
Primary: 05C48, 60G15, 60K35

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