In this talk i will present recent results obtained in collaboration with Loren Coquille (Grenoble, France) and Christof Külske (Bochum, Germany) [1,2] where we extend previous results of Gandolfo et al. [3,4] from Ising systems to general ferromagnetic finite-spin models. We focus on Gibbs measures obtained with free boundary conditions ('free states') in the very low temperature regions where we prove its non-extremality for a wide class of models including q-states Potts and clock models but also slight non-symmetric perturbations of them ('central states'). We also describe in general the non-trivial, continuous, extremal decomposition of these states at low temperature and show in particular that their decomposition into uncountably many glassy states in finite-spin models on trees is a generic phenomenon and does not rely on symmetries of the Hamiltonian.

References :

[1] L. Coquille, C. Külske, A. Le Ny. Extremal inhomogeneous Gibbs States for the SOS-models and finite-spin models on trees.Journal of Statistical Physics 191:71, 2023

[2] L. Coquille, C. Külske, A. Le Ny. Continuity of the extremal decomposition of the free state for finite-spin models on Cayler trees.ArXiv October 2023, submitted.

[3] D. Gandolfo, J. Ruiz, S. Shlosman.A manifold of pure GIbbs states of the Ising model on a Cayley tree. Journal of Statistical Physics 148:999--1005, 2012.

[4] D. Gandolfo, C. Maes, J. Ruiz, S. Shlosman. Glassy states: The free Ising model on a tree.Journal of Statistical Physics 180, no 5/6:227-237, 2020.