Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension

TL;DR

This paper studies a four-dimensional Gaussian membrane model, analyzing how a hard wall influences its maximum behavior and correlation structure, with implications for understanding critical phenomena in semiflexible membranes.

Contribution

It provides a multiscale analysis of the maximum of the Gaussian membrane model in the critical dimension, incorporating both analytic and probabilistic methods.

Findings

Characterization of the maximum of the membrane model in 4D

Description of correlation structure near the critical dimension

Insights into the effect of a hard wall on membrane fluctuations

Abstract

We consider the real-valued centered Gaussian field on the four-dimensional integer lattice, whose covariance matrix is given by the Green's function of the discrete Bilaplacian. This is interpreted as a model for a semiflexible membrane. $d=4$ is the critical dimension for this model. We discuss the effect of a hard wall on the membrane, via a multiscale analysis of the maximum of the field. We use analytic and probabilistic tools to describe the correlation structure of the field.