Maximum of the Membrane Model on Regular Trees

Abstract

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