Maximum of the membrane model on regular trees

2021/07/26 by Alessandra Cipriani, Cipriani, Alessandra, Biltu Dan +5
Mathematics · Physics and Astronomy · #60G15 #60G70 #82B20 #82B41 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2107.12276

openalex publication_date 2021/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℤ^d, 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 (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.

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