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Level-set percolation of the Gaussian free field on regular graphs I: Regular trees
JournalArticle (Originalarbeit in einer wissenschaftlichen Zeitschrift)
 
ID 4615066
Author(s) Abächerli, Angelo; Černı, Jiří
Author(s) at UniBasel Cernı, Jirí
Year 2020
Title Level-set percolation of the Gaussian free field on regular graphs I: Regular trees
Journal Electronic Journal of Probability
Volume 25
Pages / Article-Number 1-24
Abstract We study level-set percolation of the Gaussian free field on the infinite d-regular tree for fixed d >= 3. Denoting by h(*) the critical value, we obtain the following results: for h > h(*) 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(*) 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(*). Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value h(*) 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].
Publisher Institute of Mathematical Statistics and Bernoulli Society
ISSN/ISBN 1083-6489
edoc-URL https://edoc.unibas.ch/81598/
Full Text on edoc No
Digital Object Identifier DOI 10.1214/20-ejp468
ISI-Number 000541117800001
Document type (ISI) article
 
   

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