Universality of the mean-field for the Potts model

TL;DR

This paper proves that the mean-field approximation accurately predicts the log partition function for the Potts model on large, complex graphs under mild conditions, establishing universality results for various graph sequences.

Contribution

It demonstrates the asymptotic correctness of the mean-field prediction for the Potts model on a broad class of graphs, including those with negative weights and high average degree.

Findings

Mean-field prediction is asymptotically correct when trace(A_n^2)=o(n).

Universality of the limiting log partition function for ferromagnetic Potts on asymptotically regular graphs.

Large deviation principle for the empirical color measure in the Potts model.

Abstract

We consider the Potts model with $q$ colors on a sequence of weighted graphs with adjacency matrices $A_n$, allowing for both positive and negative weights. Under a mild regularity condition the mean-field prediction for the log partition function of the Potts model on a sequence of matrices $A_n$ is asymptotically correct, whenever $\text{tr}(A_n^2)=o(n)$. In particular, our results are applicable for the Ising and the Potts models on any sequence of graphs with average degree going to $+\infty$. 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.