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Critical temperature of the square lattice Potts model

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In this talk, we derive the critical temperature of the q-state Potts model on the square lattice (q \geq 2). More precisely, we consider a geometric representation of the Potts model, called the random-cluster model. Spin correlations of the Potts model get rephrased as connectivity properties of the random-cluster model. The critical temperature of the Potts model is therefore related to the critical point of the random-cluster model. For the later, a duality relation allows us to compute the critical value using a crossing estimate (similar to the celebrated Russo-Seymour-Welsh theory for percolation) and a sharp threshold theorem. This result has many applications in the eld and we will briefly mention some of them at the end of the talk. Joint work with V. Be ara.

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