Maximum of the Membrane Model on Regular Trees

Journal of Statistical Physics, vol. 190

Abstract

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 $${{\,\mathrm{{\mathbb {Z}}}\,}}^d$$ 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.

Authors 4

1. Alessandra Cipriani corresponding

   University College London · Delft University of Technology

   Affiliation as printed

   Department of Statistical Science, UCL, 1-19 Torrington Place, London, WC1E 7HB, UK

   TU Delft (DIAM), Building 28, van Mourik Broekmanweg 6, 2628 XE, Delft, The Netherlands

2. Biltu Dan

   Indian Institute of Science Bangalore

   Affiliation as printed

   Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

3. Rajat Subhra Hazra Aachen

   Leiden University · Indian Statistical Institute

   Affiliation as printed

   Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, India

   University of Leiden, Niels Bohrweg 1, 2333 CA, Leiden, The Netherlands

4. Rounak Ray

   Eindhoven University of Technology

   Affiliation as printed

   Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands