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

@article{Beffara2010TheSP,
  title={The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1},
  author={Vincent Beffara and Hugo Duminil-Copin},
  journal={Probability Theory and Related Fields},
  year={2010},
  volume={153},
  pages={511-542}
}
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 $${p_{sd}(q) = \sqrt{q} / (1+\sqrt{q})}$$. This gives a proof that the critical temperature of the q-state Potts model is equal to $${\log (1+\sqrt q)}$$ for all 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… 

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