Delocalization of two-dimensional random surfaces with hard-core constraints

TL;DR

This paper proves that two-dimensional random surfaces with hard-core constraints exhibit delocalization, with fluctuations and maximum height growing at least logarithmically with the size of the domain, using advanced probabilistic and analytical tools.

Contribution

It establishes delocalization for a broad class of random surface models with non-convex potentials, including the hammock potential, using novel adaptations of existing algorithms and reflection positivity.

Findings

Fluctuations of the surfaces are at least logarithmic in the domain size.

The expected maximum height of the surfaces grows at least logarithmically.

The results answer longstanding questions about the behavior of such models.

Abstract

We study the fluctuations of random surfaces on a two-dimensional discrete torus. The random surfaces we consider are defined via a nearest-neighbor pair potential which we require to be twice continuously differentiable on a (possibly infinite) interval and infinity outside of this interval. No convexity assumption is made and we include the case of the so-called hammock potential, when the random surface is uniformly chosen from the set of all surfaces satisfying a Lipschitz constraint. Our main result is that these surfaces delocalize, having fluctuations whose variance is at least of order $\log n$, where $n$ is the side length of the torus. We also show that the expected maximum of such surfaces is of order at least $\log n$. The main tool in our analysis is an adaptation to the lattice setting of an algorithm of Richthammer, who developed a variant of a Mermin-Wagner-type argument applicable to hard-core constraints. We rely also on the reflection positivity of the random surface model. The result answers a question mentioned by Brascamp, Lieb and Lebowitz 1975 on the hammock potential and a question of Velenik 2006.