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2020 Level-set percolation of the Gaussian free field on regular graphs I: regular trees
Angelo Abächerli, Jiří Černý
Electron. J. Probab. 25: 1-24 (2020). DOI: 10.1214/20-EJP468


We study level-set percolation of the Gaussian free field on the infinite $d$-regular tree for fixed $d\geq 3$. Denoting by $h_{\star }$ the critical value, we obtain the following results: for $h>h_{\star }$ we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level $h$; for $h<h_{\star }$ we prove that the number of vertices connected over distance $k$ above level $h$ to a fixed vertex grows exponentially in $k$ with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level $h$, at least away from the critical value $h_{\star }$. Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value $h_{\star }$ and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [1].


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Angelo Abächerli. Jiří Černý. "Level-set percolation of the Gaussian free field on regular graphs I: regular trees." Electron. J. Probab. 25 1 - 24, 2020.


Received: 11 December 2019; Accepted: 17 May 2020; Published: 2020
First available in Project Euclid: 18 June 2020

zbMATH: 1446.60079
MathSciNet: MR4115734
Digital Object Identifier: 10.1214/20-EJP468

Primary: 60G15 , 60K35

Keywords: Gaussian free field , Level-set percolation , regular tree


Vol.25 • 2020
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