Abächerli, Angelo and Černý, Jiří. (2020) Levelset percolation of the Gaussian free field on regular graphs I: Regular trees. Electronic Journal of Probability, 25. pp. 124.
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Official URL: https://edoc.unibas.ch/81598/
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Abstract
We study levelset percolation of the Gaussian free field on the infinite dregular 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 nonvanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [1].
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Wahrscheinlichkeitstheorie (Cerny) 

UniBasel Contributors:  Černý, Jiří 
Item Type:  Article, refereed 
Article Subtype:  Research Article 
Publisher:  Institute of Mathematical Statistics and Bernoulli Society 
ISSN:  10836489 
Note:  Publication type according to Uni Basel Research Database: Journal article 
Identification Number: 

Last Modified:  12 Apr 2021 13:12 
Deposited On:  12 Apr 2021 13:12 
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