Universality of the mean-field for the Potts model

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 on $$A_n$$ we show that the mean-field prediction for the log partition function is asymptotically correct, whenever $${{\mathrm{tr}}}(A_n^2)=o(n)$$ . 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.