About the slab percolation threshold for the Potts model in dimension d\ge4

Monday, 24 July, 2017

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The object of this short note is to show that the critical inverse temperature of the Potts model on \mathbb Z^3\times[-k,k]^{d-3} converges to the critical inverse temperature of the model on \mathbb Z^d. As an application, we prove that the probability that 0 is connected to distance n but not to infinity is decaying exponentially fast for the supercritical random-cluster model on \mathbb Z^d (d\ge4) associated to the Potts model. We briefly mention the connection between the present result and the classical problem of proving that the so-called slab percolation threshold coincides with the critical value for Potts models.


Hugo Duminil-Copin
Vincent Tassion