On the critical parameters of the $q\ge4$ random-cluster model on isoradial graphs

TL;DR

This paper determines the critical parameters for the $q extgreater=4$ random-cluster model on isoradial graphs using parafermionic observables, extending previous results and analyzing decay of correlations.

Contribution

It identifies the critical surface for the random-cluster model with $q extgreater=4$ on isoradial graphs and extends recent theorems to a broad class of planar graphs.

Findings

Critical surface characterized for $q extgreater=4$ on isoradial graphs.

Exponential decay of correlations in the subcritical regime.

Criticality condition for anisotropic random-cluster model on square lattice.

Abstract

The critical surface for random-cluster model with cluster-weight $q\ge 4$ on isoradial graphs is identified using parafermionic observables. Correlations are also shown to decay exponentially fast in the subcritical regime. While this result is restricted to random-cluster models with $q\ge 4$, it extends the recent theorem of the two first authors to a large class of planar graphs. In particular, the anisotropic random-cluster model on the square lattice is shown to be critical if $\frac{p_vp_h}{(1-p_v)(1-p_h)}=q$, where $p_v$ and $p_h$ denote the horizontal and vertical edge-weights respectively. We also mention consequences for Potts models.