THERMOGAMAS - Uncountably Many Series-Sinai States are Extremal on Lobachevsky Lattices (Jean Vereecke)

Talk given during the THERMOGAMAS Workshop.

Starting from the family of extremal states constructed by D'Achille-Coquille-Le Ny for the Ising model on Lobachevsky lattices (see the talk by Matteo D'Achille), I will present a way to extract from this family uncountably many ''interface states'' when the tessellation is of high degree. The latter interface states are analogues of Dobrushin states in the hyperbolic setting, i.e. weak limits of finite-volume Ising measures with boundary condition +1 on one side of a geodesics of the Poincaré disk ℍ2, and -1 on the other side. This partially solves a conjecture by Series and Sinai who proved in 1990 that any two such states are mutually singular.

The main tool we use is the Morse-Mostow lemma of hyperbolic geometry. We think there should be a (potentially stronger) alternative proof using Bowen-Series codings.