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

Abstract

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 $r\ge 3$) 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.

10 pages. The statement and the proof of the main theorem have been modified to fix a mistake in the previous version