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Title 
The selfdual point of the twodimensional randomcluster model is critical for $qgeq 1$ 

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Published in  Probability Theory and Related Fields. 2012, vol. 153, no. 3, p. 511542  
Abstract  We prove a longstanding conjecture on randomcluster models, namely that the critical point for such models with parameter $qgeq1$ on the square lattice is equal to the selfdual 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 subcritical 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  https://archiveouverte.unige.ch/unige:11981  
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arXiv: 1006.5073v1 

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