In this article we study the scaling limit of the interface model on Z d where the Hamiltonian is given by a mixed gradient and Laplacian interaction. We show that in any dimension the scaling limit is given by the Gaussian free field. We discuss the appropriate spaces in which the convergence takes place. While in infinite volume the proof is based on Fourier analytic methods, in finite volume we rely on some discrete PDE techniques involving finite-difference approximation of elliptic boundary value problems.2000 Mathematics Subject Classification. 31B30, 60J45, 60G15, 82C20. Key words and phrases. Mixed model, Gaussian free field, membrane model, random interface, scaling limit.RSH acknowledges MATRICS grant from SERB and the Dutch stochastics cluster STAR (Stochastics Theoretical and Applied Research) for an invitation to TU Delft where part of this work was carried out. The authors thank Francesco Caravenna for helpful discussions.
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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