Dynamical phase transition for the homogeneous multi-component Curie-Weiss-Potts model

In this paper, we study the homogeneous multi-component Curie-Weiss-Potts model with $q \geq 3$ spins. The model is defined on the complete graph $K_{Nm}$, whose vertex set is equally partitioned into $m$ components of size $N$. For a configuration $\sigma: \{1, \cdots, Nm\} \to \{1, \cdots, q\},$ the Gibbs measure is defined by $$\mu_{N,\beta}(\sigma) =\frac{1}{Z_{N,\beta}} \exp\Big(\frac{\beta}{N} \sum_{v,w=1}^{Nm}\mathcal{J}(v,w)\, \mathbb{1}_{\{\sigma(v)=\sigma(w)\}}\Big),$$ where $Z_{N, \beta}$ is a normalizing constant, and $\beta>0$ is the inverse temperature parameter. The interaction coefficients are $\mathcal{J}(v, w) = \frac{J}{1 + (m-1) \lambda}$, for $v, w$ in the same component, and $\mathcal{J}(v, w) = \frac{J \lambda}{1 + (m-1)\lambda}$ for $v, w$ in the different components, where $\lambda \in (0, 1)$ is the relative strength of inter-component interaction to intra-component interaction, and $J>0$ is the effective interaction strength. We identify a dynamical phase transition at the critical inverse temperature $\beta_{\operatorname{cr}} = \beta_{s}(q)/J$, where $\beta_{s}(q)$ is maximal inverse temperature guaranteeing a unique critical point of the free energy in the Curie-Weiss-Potts model arXiv:1204.4503. By extending the aggregate path method arXiv:1312.6728 to our multi-component setting, we prove $O(N \log N)$ mixing time in the high-temperature regime $\beta<\beta_{s}(q)/J.$ In the low-temperature regime $\beta > \beta_{s}(q)/J,$ we further show exponential mixing time by a metastability. This is the first result for the dynamical phase transition in the multi-component Potts model.