A note on Schramm's locality conjecture for random-cluster models

In this note, we discuss a generalization of Schramm’s locality conjecture to the case of random-cluster models. We give some partial (modest) answers, and present several related open questions. Our main result is to show that the critical inverse temperature of the Potts model on $\mathbb Z^r\times(\mathbb Z/2n\mathbb Z)^{d-r}$(with$r\ge3$) converges to the critical inverse temperature of the model on ${\mathbb{Z}}^d$ as $n$ tends to infinity. Our proof relies on the infrared bound and, contrary to the corresponding statement for Bernoulli percolation, does not involve renormalization arguments.