Abächerli, Angelo and Černý, Jiří. (2020) Levelset percolation of the Gaussian free field on regular graphs II: Finite expanders. Electronic Journal of Probability, 25. pp. 139.
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Official URL: https://edoc.unibas.ch/81599/
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Abstract
We consider the zeroaverage Gaussian free field on a certain class of finite dregular graphs for fixed d >= 3. This class includes dregular expanders of large girth and typical realisations of random dregular graphs. We show that the level set of the zeroaverage Gaussian free field above level h(*), exhibits a phase transition at level which agrees with the critical value for levelset percolation of the Gaussian free field on the infinite dregular 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(*), and on the contrary, whenever h < h(*), 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(*), 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].
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 13:08 
Deposited On:  12 Apr 2021 13:08 
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