Maximum of the membrane model on regular trees

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.