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Maximum of the Membrane Model on Regular Trees

Cipriani, A; Dan, B; Hazra, RS; Ray, R; (2023) Maximum of the Membrane Model on Regular Trees. Journal of Statistical Physics , 190 (1) , Article 25. 10.1007/s10955-022-03043-w. Green open access

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Abstract

The discrete membrane model is a Gaussian random interface whose inverse covariance is given by the discrete biharmonic operator on a graph. In literature almost all works have considered the field as indexed over Zd, and this enabled one to study the model using methods from partial differential equations. In this article we would like to investigate the dependence of the membrane model on a different geometry, namely trees. The covariance is expressed via a random walk representation which was first determined by Vanderbei in (Ann Probab 12:311–314, 1984). We exploit this representation on m-regular trees and show that the infinite volume limit on the infinite tree exists when m≥ 3. Further we determine the behavior of the maximum under the infinite and finite volume measures.

Type: Article
Title: Maximum of the Membrane Model on Regular Trees
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s10955-022-03043-w
Publisher version: https://doi.org/10.1007/s10955-022-03043-w
Language: English
Additional information: © 2022 Springer Nature Switzerland AG. This article is licensed under a Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).
Keywords: Random interfaces, Membrane model, Trees, Extremes, Random walk representation
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery-pp.ucl.ac.uk/id/eprint/10161755
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