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

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Published in Probability Theory and Related Fields. 2012, vol. 153, no. 3, p. 511-542
Abstract We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter $qgeq1$ on the square lattice is equal to the self-dual point $p_{sd}(q) = sqrt q /(1+sqrt q)$. This gives a proof that the critical temperature of the $q$-state Potts model is equal to $log (1+sqrt q)$ for all $qgeq 2$. 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 $qgeq 1$, in contrast to earlier methods valid only for certain given $q$. The proof extends to the triangular and the hexagonal lattices as well.
Stable URL http://archive-ouverte.unige.ch/unige:11981
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Deposited on : 2010-10-01

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