Limit shapes in the 6-Vertex Model
Lundi 28/03/2011, 11:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
In 2-dimensional Statistical Mechanics, models like Dimer Coverings on bipartite graphs or the 6-Vertex Model may show anisotropy and strong dependence from the boundary conditions (at difference, e.g., with Ising, Potts or 8-Vertex Model). When the anisotropy is such that there are ``frozen regions'', interface curves arise, called ``Arctic curves''. Dimer Models are fermionic, and their Arctic-curve phenomenology is now well understood, after the work of Kenyon, Okounkov and Sheffield. For the 6-Vertex Model out of the fermionic point, one only has the comparatively weaker tools of Integrable Systems: the arctic curve is known only for the square, after the work of Colomo and A. Pronko, later joined by P.Zinn-Justin. We propose an alternate heuristic derivation of the Colomo-Pronko result, using a geometric method, and avoiding the detour on Emptiness Formation Probability and Random Matrix Theory (together with the corresponding non-rigorous steps). We argue that the new method is quite flexible, and, as an illustration, we derive a new arctic curve, for the 6-Vertex Model on a 3-bundle domain. Work in collaboration with Filippo Colomo.