On fluctuations of random surfaces

Random surfaces in statistical physics are commonly modeled by a real-valued function phi on the vertex set of a lattice-like graph, which is chosen at random while the underlying distribution penalizes large gradients of the function between adjacent vertices. Precisely, given an even function V, termed the potential, the energy H(phi) is computed as the sum of V over the nearest-neighbor gradients of phi, and the probability density of phi is set proportional to exp(-H(phi)). The most-studied case is when V is quadratic, resulting in the so-called discrete Gaussian free field. For what follows, we set the graph to be an even torus of size 2L an dimension 3 or higher, and, fixing a vertex v_0, apply a simple boundary condition phi(v_0) = 0. An application of the Brascamp-Lieb inequality yields that when the potential is uniformly convex the variance Var(phi(v)) is uniformly bounded from above, independently of the vertex v and the size 2L of the torus. We show that the same conclusion still holds with slightly relaxed assumptions on V, namely, when V is convex and the inequality V"(x) > 0 holds for Lebesgue-a.e. value of x. This is a joint work with Ron Peled.
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https://zoom.us/j/93175142429?pwd=VDViRHNOSlZSVUM5ZU03SGZyZy8xQT09
Id: 931-7514-2429 passw=057376
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