Universality of the mean-field for the Potts model

• 成果类型: Article
• 署名作者: Basak, Anirban; Mukherjee, Sumit
• 署名单位: Duke University; Columbia University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0718-0
• 发表日期: 2017
• 页码: 557-600
• 关键词: phase-transitions; LIMIT-THEOREMS; ising-models; graphs

摘要

We consider the Potts model with q colors on a sequence of weighted graphs with adjacency matrices , allowing for both positive and negative weights. Under a mild regularity condition on we show that the mean-field prediction for the log partition function is asymptotically correct, whenever . In particular, our results are applicable for the Ising and the Potts models on any sequence of graphs with average degree going to . Using this, we establish the universality of the limiting log partition function of the ferromagnetic Potts model for a sequence of asymptotically regular graphs, and that of the Ising model for bi-regular bipartite graphs in both ferromagnetic and anti-ferromagnetic domain. We also derive a large deviation principle for the empirical measure of the colors for the Potts model on asymptotically regular graphs.