Mixing phases and metastability for the Glauber dynamics on the p-spin Curie-Weiss model

The Glauber dynamics for the classical 2-spin Curie-Weiss model on N nodes with inverse temperature β and zero external field is known to mix in time Θ(NlogN) for β<12, in time Θ(N3∕2) at β=12, and in time exp(Ω(N)) for β>12. In this paper, we consider the p-spin generalization of the Curie-Weiss model with an external field h, and identify three disjoint regions partitioning the parameter space, with the corresponding Glauber dynamics exhibiting three different orders of mixing times in these regions. The construction of these disjoint regions depends on the number of local maximizers of a certain negative free-energy function Hβ,h,p, and the behavior of the second derivative of Hβ,h,p at such a local maximizer. Specifically, we show that if Hβ,h,p has a unique local maximizer m∗ with Hβ,h,p″(m∗)<0 and no other stationary point, then the Glauber dynamics mixes in time Θ(NlogN); if Hβ,h,p has multiple local maximizers, then the mixing time is exp(Ω(N)); and if Hβ,h,p has a unique local maximizer m∗ with Hβ,h,p″(m∗)=0, then the mixing time is Θ(N3∕2). We provide an explicit description of the geometry of these three different phases within the parameter space. Finally, we show that if Hβ,h,p has multiple local maximizers (metastable states), then one can create a restricted version of the original Glauber dynamics, which still mixes in time Θ(NlogN).