Correlations in random cluster model at $q=1$

TL;DR

This paper derives a combinatorial formula for correlations in the random cluster model at q=1, clarifying the nature of element interactions in this specific case.

Contribution

It provides the first explicit combinatorial formula for correlations in the random cluster model at q=1, extending understanding beyond known cases.

Findings

Formula for correlation difference at q=1 derived

Clarifies negative/positive correlation behavior at q=1

Connects to uniform spanning tree case at q=0

Abstract

Let $\mu$ be a measure that samples a subset of a finite ground set, and let $\mathcal{A}_e$ be the event that element $e$ is sampled. The measure $\mu$ is negatively correlated if for any pair of elements $e, f$ one has $\mu(\mathcal{A}_e \cap \mathcal{A}_f) - \mu(\mathcal{A}_e) \mu(\mathcal{A}_f) \leq 0$. A measure is positively correlated if the direction of the inequality is reversed. For the random cluster model on graphs positive correlation between edges is known for $q \geq 1$ due to the FKG inequality, while the negative correlation is only conjectured for $0 \leq q \leq 1$. The main result of this paper is to give a combinatorial formula for the difference in question at $q=1$. Previously, such a formula was known in the uniform spanning tree case, which is a limit of the random cluster model at $q=0$.