The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1

By Vincent Beffara, H. Duminil-Copin. Probability theory and related fields, 2010

226 citations recorded by the source corpus.

We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter q ≥ 1 on the square lattice is equal to the self-dual point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p_{sd}(q) = \sqrt{q} / (1+\sqrt{q})}$$\end{document}. This gives a proof that the critical temperature of the q-state Potts model is equal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\log (1+\sqrt q)}$$\end{document} for all q ≥ 2. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all q ≥ 1, in contrast to earlier methods valid only for certain given q. The proof extends to the triangular and the hexagonal lattices as well.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0b47c43e236e4967940638b1739797f5a6da0e24

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In Condensed Matter Physics

1. Aran Raoufi
2. H. Duminil-Copin
3. Vincent Tassion
4. Michael Aizenman

In Mathematical Physics

1. Aran Raoufi
2. H. Duminil-Copin
3. Vincent Tassion
4. Michael Aizenman

In Statistics and Probability

1. Aran Raoufi
2. H. Duminil-Copin
3. Vincent Tassion
4. Michael Aizenman

Citations in the corpus

• Sharp phase transition for the random-cluster and Potts models via decision trees (2017, background): “This result was originally proved in [BD12], where exponential decay of correlations below pc is proved using Russo-Seymour-Welsh type arguments and a generalization [GG06] of the KKL result [KKL88, BKK92].”