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

TL;DR

This paper explores a generalization of Schramm's locality conjecture for random-cluster models, providing partial answers and showing convergence of critical inverse temperatures in certain lattice structures without using renormalization.

Contribution

It extends Schramm's locality conjecture to random-cluster models and proves convergence of critical inverse temperatures on specific lattice sequences.

Findings

Critical inverse temperature on $ imes( ext{Z/2nZ})^{d-r}$ converges to that on $ ext{Z}^d$ as n→∞

Proof uses infrared bound without renormalization

Provides partial answers and open questions on the conjecture

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 $r\ge3$) 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.