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The membrane model (MM) is a random interface model for separating surfaces that tend to preserve curvature. It is a Gaussian interface whose inverse covariance is given by the discrete biharmonic operator. It is a very close relative of the discrete Gaussian free field, for which the inverse covariance is given by the discrete harmonic operator. We consider the MM on the d-dimensional integer lattice. We study its scaling limit using some discrete PDE techniques involving finite difference approximation of elliptic boundary value problems. Also, we discuss the behavior of the maximum of the model. Then we consider the MM on regular trees and investigate a random walk representation for the covariance. Exploiting the random walk representation for the covariance, we determine the behavior of the maximum of the MM on regular trees.
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