Random Lipschitz functions and 6-vertex model via spin representations (Alexander Glazman, Tel Aviv)

24.09.2018 15:15 – 16:15

In this talk by a Lipschitz function we mean an assignment of integer values to each vertex of the triangular lattice, so that the difference between the values at any two adjacent vertices is at most 1. Pick such a function uniformly among those with values 0 outside some large but finite domain. We show that as the size of the domain tends to infinity, the fluctuations of such function are of logarithmic order and at each scale with positive probability there is a level-line (RSW result). The set of level lines of Lipschitz functions is distributed according to the loop O(2) model with x=1. Our results imply that the latter has a unique Gibbs measure.

The 6-vertex model is supported on arrow-configurations on edges of Z^2 that satisfy the ice-rule: each vertex has two incoming and two outgoing edges. Given a positive parameter c, the probability of each configuration is proportional to c to the number of vertices having either two vertical or two horizontal incoming edges. In this talk, we show that for c > 2 there exist four distinct Gibbs measures that are extreme, invariant under parity-preserving translations, and can be obtained as thermodynamical limits under specific boundary conditions. In contrast, for c = 2, these thermodynamical limits coincide and give a Gibbs measure that is extreme and invariant under all translations. We will also discuss the variance of the associated height function: we show that it is finite when c > 2 and logarithmic at c=2.

At the basis of both these studies is a specific two-colour spin representation that satisfies the FKG inequality.

Based on joint works with Ioan Manolescu and Ron Peled.

Lieu

Room 17, Séminaire "Mathématique Physique"

entrée libre