2022/11/21 by Kyunghoo Mun, Mun, Kyunghoo
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2211.11463
openalex publication_date 2022/11/21 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/28
In this paper, we study the homogeneous multi-component Curie-Weiss-Potts model with q ≥ 3 spins. The model is defined on the complete graph KNm, whose vertex set is equally partitioned into m components of size N. For a configuration σ: \1, ⋯, Nm\ → \1, ⋯, q\, the Gibbs measure is defined by μN,β(σ) =\frac1ZN,β exp(\fracβN ∑v,w=1NmJ(v,w) \mathbb1_\σ(v)=σ(w)\), where ZN, β is a normalizing constant, and β>0 is the inverse temperature parameter. The interaction coefficients are J(v, w) = (J)/(1 + (m-1) λ), for v, w in the same component, and J(v, w) = (J λ)/(1 + (m-1)λ) for v, w in the different components, where λ∈ (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 βcr = βs(q)/J, where βs(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 β<βs(q)/J. In the low-temperature regime β> βs(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.