Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$

TL;DR

This paper proves that for q>4, the planar random-cluster and Potts models exhibit a discontinuous phase transition, with multiple infinite-volume measures and exponential decay of correlations, using transfer matrix eigenvalue analysis.

Contribution

It provides a rigorous proof of discontinuous phase transitions for q>4 in the planar random-cluster and Potts models, including explicit correlation length calculations.

Findings

Multiple infinite-volume measures at criticality

Exponential decay of correlations with free boundary conditions

Correlation length behaves as exp(π^2/√(q-4)) as q approaches 4

Abstract

We prove that the $q$-state Potts model and the random-cluster model with cluster weight $q>4$ undergo a discontinuous phase transition on the square lattice. More precisely, we show - Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, - Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and - Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models. The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model. As a byproduct, we rigorously compute the correlation lengths of the critical random-cluster and Potts models, and show that they behave as $\exp(\pi^2/\sqrt{q-4})$ as $q$ tends to 4.