In this paper, we study the homogeneous multi-component Curie-Weiss-Potts model with \(q \ge 3\) 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 \(\sigma : \{1, \cdots , Nm\} \rightarrow \{1, \cdots , q\},\) the Gibbs measure is defined by \( \mu _{N, \beta }(\sigma ) = {1 \over Z_{N, \beta }} \exp \left( {\beta \over N} \sum _{v, w =1}^{Nm} \mathcal {J}(v, w) \mathbbm {1}\{ \sigma (v) = \sigma (w)\}\right) , \) where \(Z_{N, \beta }\) is the normalizing constant, and \(\beta >0\) is the inverse-temperature parameter. The interaction coefficient is \( \mathcal {J}(v, w) = {\left\{ \begin{array}{ll} \frac{1}{1+(m-1)J} & \text {if } v, w \text { are in the same component,}\\ \frac{J}{1+(m-1)J} & \text {if } v, w \text { are in different components,} \end{array}\right. } \) where \(J \in (0, 1)\) is the relative strength of inter-component interaction to intra-component interaction. We identify a dynamical phase transition at the critical inverse-temperature \(\beta _{s}(q)\) , which is the same threshold as for the one-component Potts model [5] and depends only on the number of spins q, but is independent of the number of components m and relative interaction strength \(J \in (0, 1).\) By extending the aggregate path method [19] to multi-component setting, we prove that the mixing time is \(O(N \log N)\) in the subcritical regime \(\beta <\beta _{s}(q).\) In the supercritical regime \(\beta > \beta _{s}(q),\) we further show that the mixing time is exponential in N via a metastability analysis. This is the first result for the dynamical phase transition in the multi-component Potts model.
