Sharp asymptotics of correlation functions in the subcritical long-range random-cluster and Potts models

TL;DR

This paper derives sharp asymptotic formulas for connection probabilities in subcritical long-range random-cluster and Potts models, revealing precise decay rates and dependencies.

Contribution

It introduces a method to obtain asymptotics of connection probabilities in random-cluster models with long-range interactions, applicable to models with one-monotonic representations.

Findings

Connection probability asymptotics match the form rac{1}{q}hi(eta)^2eta J_{0,x}

Method applies broadly to spin models with suitable random-cluster representations

Results provide detailed decay rates in subcritical long-range models.

Abstract

For a family of random-cluster models with cluster weights $q\geq 1$, we prove that the probability that $0$ is connected to $x$ is asymptotically equal to $\tfrac{1}{q}\chi(\beta)^{2}\beta J_{0,x}$. The method developed in this article can be applied to any spin model for which there exists a random-cluster representation which is one-monotonic.