THERMOGAMAS - Uncountably Many Extremal Ising Gibbs States on Lobachevsky Lattices (Matteo d'Achille)

Abstract: Aizenman and Higuchi famously proved that, at low temperatures, any Gibbs state of the Ising model on ℤ² is a convex combination of two extremal states.

In this talk I will exhibit uncountably many extremal low-temperature Gibbs states for the Ising model on the graph given by a regular tessellation of the hyperbolic plane (a.k.a. Lobachevsky lattice).

The proof combines two main ingredients:

- An excess energy lemma (which holds for the Ising model defined on any non-amenable graph) providing a uniform lower bound to the cost of a spin flip via a linear combination of the number of frustrated bonds and of the Cheeger constant of the graph;

- A layer decomposition specific to these lattices, due to Rietman–Nienhuis–Oitmaa and Moran, which allows us to build Dobrushin-like interfaces by a suitable gluing of two infinite trees in the dual graph. En passant, I will also prove that certain "regular balls" built via this layer decomposition solve the isoperimetric problem at fixed volume.

This partially solves a conjecture of Series-Sinai.

Talk mostly based on 10.1214/25-ECP724 in collaboration with Loren Coquille (Grenoble) and Arnaud Le Ny (Paris-Est Créteil); and on 2504.14080 (to appear) with Vanessa Jacquier (Padova) and Wioletta M. Ruszel (Utrecht).