Spinodal Points for Potts Model on the Bethe Lattice

The main interest of the present paper is to discuss the ferromagnetic Potts model that has only a spatially ferromagnetic uniform interaction on random regular graphs (Bethe lattice) and to give perturbative analyses around supercooling spinodal points under the first order phase transition. It is known that the ferromagnetic Potts model on regular tree graphs have the first order phase transition and appear not only in equilibrium states but also local maximum states and local minimum states and that such local minimum and local maximum states correspond to metastable and unstable states, respectively. In some statistical inferences, statistical machine learning paths often go through near not only equilibrium states but also metastable states and unstable states and we have to compute some statistical quantities in such states. Hence we have to clarify detailed behaviours of statistical quantities in treated prior probabilistic models as fundamental knowledge in the statistical mechanical point of view. And it is expected that there exist rich expressive abilities as metastable states in such prior probabilistic models. Behaviours of statistical quantities in metastable states and unstable states are clarified not only numerically but also theoretically in the Potts model on the Bethe lattice for realizing rich expressive, high speed and high performance statistical machine learning frameworks.