Abächerli, Angelo and Černý, Jiří. (2019) Levelset percolation of the Gaussian free field on regular graphs II: Finite expanders. Preprints Fachbereich Mathematik, 2019 (15).

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Abstract
We consider the zeroaverage Gaussian free field on a certain class of finite $d$regular graphs for fixed $d\geq 3$. This class includes $d$regular expanders of large girth and typical realisations of random $d$regular graphs. We show that the level set of the zeroaverage Gaussian free field above level $h$ exhibits a phase transition at level $h_\star$, which agrees with the critical value for levelset percolation of the Gaussian free field on the infinite $d$regular 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 $h>h_\star$, and on the contrary, whenever $h<h_\star$, a 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 $h<h_\star$, 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 [AC1].
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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