Conférence annuelle du GDR branchement

Uncountably many extremal inhomogeneous states for the Ising model on regular tilings of the hyperbolic plane

29 janv. 2025, 10:45

45m

Orsay

Orateur

Matteo D'Achille (Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay)

Description

Series-Sinai have shown in the nineties that the ferromagnetic n.n. Ising model defined on the Cayley graph of a co-compact group of isometries of the hyperbolic plane ${\mathbb{H}}_2$ exhibits uncountably many, mutually singular Gibbs states at very low temperature ---one for every bi-infinite geodesic of ${\mathbb{H}}_2$.

They also conjectured the extremality of their states but the problem has been open ever since.

In this talk I will prove the existence of uncountably many extremal inhomogeneous Gibbs states for the Ising model on regular tilings of ${\mathbb{H}}_2$. I will also prove a refined Peierls bound for the critical temperature and sketch a few research directions.

Joint work with Loren Coquille (Institut Fourier, Grenoble) and Arnaud Le Ny (LAMA, Université Paris-Est Créteil).