Agenda

Résumé : Below the so-called spin-glass temperature, the free state of the Ising model on a regular tree is not extremal. Moreover, its extremal decomposition is a continuous measure, supported on (uncountably many) inhomogeneous extremal states, which have some kind of « glassy » feature. I will present some ideas behind a proof of this result, which provides explicit concentration bounds on ”branch overlaps”, playing the role of an order parameter for typical extremals. The proof extends quite generally to ferromagnetic finite-spin models, even slightly asymmetric (i.e. where small inhomogeneous field terms are added), which shows that the above behaviour is generic on regular trees. This is a joint work with Christof Külske (Bochum) and Arnaud Le Ny (Paris-Est Créteil).