On the membrane model and the discrete Bilaplacian

Abstract

This thesis is concerned with the membrane model, an example of a discrete random interface model. This model arises, for example, when studying thermal fluctuations in biomembranes. Mathematically, it is given as a Gibbs measure with a Hamiltonian that involves squared second derivatives. The membrane model is a natural variant of the discrete Gaussian free field, the best-known example of a random interface model. While the two models are expected to behave similarly, the membrane model lacks some of the useful features of the latter model, and so new techniques are required to study it. <br /> The starting point is that the covariance of the membrane model is given as the Green's function of the discrete Bilaplacian, and so a second topic of this thesis is the study of this Green's function and the associated discrete differential operator. For this purpose we employ various techniques from numerical analysis and the theory of partial differential equations. <br /> Our results on the membrane model concern its extrema, and the phenomena of entropic repulsion and pinning. We study its maximum in the critical dimension 4 and prove that it converges in distribution, when recentred suitably. We also consider the effect of a hard wall on the field, which is related to entropic repulsion, and in the subcritical dimensions 2 and 3 we establish asymptotics for the probability of the interface to stay above the wall. Furthermore, we investigate pinning, that is, the localizing effect that a small attractive potential has on the interface. In the critical and supercritical dimensions 4 and above we establish precise asymptotics on the variances and covariances of the pinned field. <br /> An essential part of the proofs of these results are estimates for the Green's function of the discrete Bilaplacian. In particular, we establish new estimates for this Green's function in dimensions 2, 3 and 4. In dimensions 2 and 3 our approach is based on compactness arguments and results for continuous partial differential equations in domains with singularities. In dimension 4 we use a different approach and apply estimates for the approximation quality of finite difference schemes for the Bilaplacian. <br /> We also elaborate on those estimates beyond what would be needed for the application to the Green's function, and establish a more general improved error estimate for such finite difference schemes.