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

Hugo Duminil-Copin, Vincent Tassion

Published 2017-07-24Version 1

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.