Delocalization of Two-Dimensional Random Surfaces with Hard-Core Constraints

Abstract

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 et al. on the hammock potential and a question of Velenik.

Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations

“…The case of real-valued height functions is better understood. In particular, convergence to the GFF was established for uniformly convex symmetric potentials (under additional regularity assumptions) [29] and the delocalisation was proven for some nonconvex nearest-neighbour potentials [30]. (2) model.…”

Power-law decay of weights and recurrence of the two-dimensional VRJP

“…Second, the Mermin-Wagner method is easiest to implement for gradient fields whose interaction function U is twice-continuously differentiable with sup U < ∞ (see, e.g., [15, § 1.1] or [20, § 2.6]). As the hyperbolic cosine function does not satisfy this bound we need to resort to a more sophisticated version of the argument, based on ideas of Richthammer [21] and developed in [15] (see the "addition algorithm" of § 4). A price to pay is that an additional a priori input is required: We need to show that for some sufficiently large constant K, the random set of edges on which the gradient of u exceeds K in absolute value is sparse in an appropriate probabilistic sense.…”

“…A price to pay is that an additional a priori input is required: We need to show that for some sufficiently large constant K, the random set of edges on which the gradient of u exceeds K in absolute value is sparse in an appropriate probabilistic sense. For other models, such an input was established either using reflection positivity [15] or using symmetries of the state space [19]. Here, this input is proved by the approach of [1], namely, by considering together the VRJP and RWRE pictures for the field u (see § 3).…”

“…The final ingredient used in our proof is the following, so called addition algorithm, which is introduced in [15] following earlier work of Richthammer [21].…”

Lectures on the Spin and Loop O(n) Models

“…The idea to combine the approaches is introduced in a forthcoming paper of Gagnebin, Mi loś and Peled [56], where it is pushed further to prove power-law decay of correlations for any measurable potential U satisfying only very mild integrability assumptions. The work [56] relies further on ideas used by Ioffe-Shlosman-Velenik [72], Richthammer [102] and Mi loś-Peled [91].…”