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 ≥ 3 q ≥ 3 spins. The model is defined on the complete graph KNm K Nm , whose vertex set is equally partitioned into m components of size N . For a configuration σ : \1, ⋯ , Nm\ → \1, ⋯ , q\, σ : { 1 , ⋯ , N m } → { 1 , ⋯ , q } , the Gibbs measure is defined by μ N, β(σ ) = 1 ZN, β exp ( β N ∑ v, w =1Nm J(v, w) 1\ σ (v) = σ (w)\) , μ N , β ( σ ) = 1 Z N , β exp β N ∑ v , w = 1 Nm J ( v , w ) 1 { σ ( v ) = σ ( w ) } , where ZN, β Z N , β is the normalizing constant, and β >0 β > 0 is the inverse-temperature parameter. The interaction coefficient is $$ J(v, w) = {\{ {array}{ll} 1/1+(m-1)J & if v, w are in the same component,\\ J/1+(m-1)J & if v, w are in different components, {array}. } $$ J ( v , w ) = 1 1 +