Planar random-cluster model: scaling relations

TL;DR

This paper establishes scaling relations among critical exponents for the planar random-cluster model on or q, using novel coupling techniques and stability of crossing probabilities, generalizing classical percolation results.

Contribution

It introduces new coupling methods and interpretations of influence to derive scaling relations and stability results for the random-cluster model.

Findings

Derived scaling relations between critical exponents.

Proved stability of crossing probabilities in near-critical regimes.

Generalized Kesten's scaling relation involving mixing rate exponent.

Abstract

This paper studies the critical and near-critical regimes of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in[1,4]$ using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents $\beta$, $\gamma$, $\delta$, $\eta$, $\nu$, $\zeta$ as well as $\alpha$ (when $\alpha\ge0$). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalization of Kesten's classical scaling relation for Bernoulli percolation involving the ``mixing rate'' critical exponent $\iota$ replacing the four-arm event exponent $\xi_4$.