Abächerli, Angelo and Černý, Jiří. (2019) Levelset percolation of the Gaussian free field on regular graphs I: Regular trees. Preprints Fachbereich Mathematik, 2019 (14).

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Abstract
We study levelset 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 nonvanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [AC19].
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Wahrscheinlichkeitstheorie (Cerny) 12 Special Collections > Preprints Fachbereich Mathematik 

UniBasel Contributors:  Černý, Jiří 
Item Type:  Preprint 
Publisher:  Universität Basel 
Language:  English 
Last Modified:  07 Aug 2020 10:37 
Deposited On:  07 Aug 2020 10:37 
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