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Universality of the mean-field for 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 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.

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Acknowledgments

We thank Andrea Montanari for suggesting to look at the Ising measure on hypercube, Sourav Chatterjee for pointing out the reference [16], and Amir Dembo for helpful comments on earlier version of the manuscript. We also thank Sourav Chatterjee, Amir Dembo, and Andrea Montanari for many helpful discussions. We further thank Marek Biskup and Aernout Van Enter for pointing out the references [6] and [13] respectively. We are grateful to two anonymous referees for their detailed comments and suggestions which have improved the quality of this paper.

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Correspondence to Anirban Basak.

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Basak, A., Mukherjee, S. Universality of the mean-field for the Potts model. Probab. Theory Relat. Fields 168, 557–600 (2017). https://doi.org/10.1007/s00440-016-0718-0

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  • DOI: https://doi.org/10.1007/s00440-016-0718-0

Keywords

  • Ising measure
  • Potts model
  • Log partition function
  • Mean-field
  • Large deviation

Mathematics Subject Classification

  • 60K35
  • 82B20
  • 82B44