Universality of the mean-field for the Potts model

2015/08/17 by Anirban Basak, Basak, Anirban, Sumit Mukherjee +1 · 4 citations
Mathematics · Physics and Astronomy · #60K35 #82B20 #82B44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60K35 #msc:82B20 #msc:82B44

paper · pdf · doi:10.48550/arxiv.1508.03949

35 pages, minor chages, to appear in PTRF

arxiv created 2016/05/05 · arxiv updated 2016/05/06

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 tr(A_n²)=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 +∞. 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.

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• Asymptotic Normality of Subgraph Counts in Sparse Inhomogeneous Random Graphs
• Causal inference under interference: computational barriers and algorithmic solutions
• From Ising to Potts: Physics-inspired Potts machines of coupled oscillators for low-energy sampling and combinatorial optimization

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