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.
Highlights
In this paper we study the fluctuations of random surface models in two dimensions
In this paper we prove a lower bound of order log n on the variance for a wide class of potentials, which includes the hammock potential
In the lemma we investigate the gradient of T +(φ), establishing property (3) from Sect. 2.1 for T +
Summary
In this paper we study the fluctuations of random surface models in two dimensions. We consider the following family of models. It is expected that this variance is of order log n under mild conditions on U This has been shown when the potential U is twice continuously differentiable with U bounded away from zero and infinity, and certain extensions of this class, as discussed in the survey paper [31, Remarks 6 and 7]. By E Both (φ) the random subgraph of T2n difficulties described above may consisting of be overcome all by showing that with high probability, the subgraph E(φ) is “subcritical” in the sense that its connected components are small Proving this turns out to be a non-trivial task, which requires us to make use of reflection positivity techniques, the chessboard estimate. 3 we discuss reflection positivity for random surface models and prove, via the chessboard estimate, that the subgraph of edges with extremal slopes mentioned in the previous section is “subcritical” with high probability.
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