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.
Categories: math.PR cond-mat.stat-mech math-ph math.MP