Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with $${1 \le q \le 4}$$ 1 ≤ q ≤ 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 1. Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, 2. Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and 3. 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 randomcluster and Potts models, and show that they behave as exp(π 2 √ q − 4) as q tends to 4.Section 4: Fourier computations. The study will require certain computations using Fourier decompositions. While these computations are elementary, they may be lengthy, and would break the pace of the proofs. We therefore defer all of them to Section 4.Notation. Most functions hereafter depend on the parameter ∆ = 2−c 2 2 < −1. For ease of notation, we will generally drop the dependency in ∆, and recall it only when it is relevant. We write ∂ i for the partial derivative in the i th coordinate.

Unusual changeover in the transition nature of local-interaction Potts models

“…Our results for q c , in particular that of the honeycomb lattice, stand in contrast to the well known result q c = 4 of the usual model [4,5,7,8].…”

“…His findings were believed to be lattice independent [6]. Recently, Duminil-Copin et al [7,8] have rigorously confirmed Baxter's predictions using the random cluster representation [9] of the nearest-neighbor interaction model.…”

Critical Percolation and the Minimal Spanning Tree in Slabs

“…The proof relies on a result of Russo, Seymour, and Welsh [23,24] that relates crossing probabilities for rectangles with different aspect ratios. Recently the boxcrossing property has been extended to planar percolation processes with spatial dependencies, e.g., continuum percolation [1,26] or the random-cluster model [10].…”

Absence of Infinite Cluster for Critical Bernoulli Percolation on Slabs

“…On Z 2 , the continuity of the phase transition was recently proven [DCST14] for dependent percolation models known as random-cluster models with cluster-weight q ∈ [1, 4] (the special case q = 1 corresponds to Bernoulli percolation). The continuity of the phase transition for q = 1 and 2 was previously established by Harris [Har60] and Onsager [Ons44] respectively.…”