Maximum of the membrane model on regular trees

TL;DR

This paper investigates the discrete membrane model on regular trees, establishing the existence of the infinite volume limit and analyzing the behavior of the maximum of the field in this new geometric setting.

Contribution

It extends the study of the membrane model from lattice structures to regular trees, providing new insights into its covariance and maximum behavior on such graphs.

Findings

Infinite volume limit exists for m-regular trees with m ≥ 3

Covariance expressed via a random walk representation on trees

Behavior of the maximum analyzed under infinite and finite volume measures

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 $\mathbb{Z}^d$, 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 $m\ge 3$. Further we determine the behavior of the maximum under the infinite and finite volume measures.