Open Access
2023 On the transition between the disordered and antiferroelectric phases of the 6-vertex model
Alexander Glazman, Ron Peled
Author Affiliations +
Electron. J. Probab. 28: 1-53 (2023). DOI: 10.1214/23-EJP980


The symmetric six-vertex model with parameters a,b,c>0 is expected to exhibit different behavior in the regimes a+b<c (antiferroelectric), |ab|<ca+b (disordered) and |ab|>c (ferroelectric). In this work, we study the way in which the transition between the regimes a+b=c and a+b<c manifests.

When a+b<c, we show that the associated height function is localized and its extremal periodic Gibbs states can be parametrized by the integers in such a way that, in the n-th state, the heights n and n+1 percolate while the connected components of their complement have diameters with exponentially decaying tails. When a+b=c, the height function is delocalized.

The proofs rely on the Baxter–Kelland–Wu coupling between the six-vertex and the random-cluster models and on recent results for the latter. An interpolation between free and wired boundary conditions is introduced by modifying cluster weights. Using triangular lattice contours (T-circuits), we describe another coupling for height functions that in particular leads to a novel proof of the delocalization at a=b=c.

Finally, we highlight a spin representation of the six-vertex model and obtain a coupling of it to the Ashkin–Teller model on Z2 at its self-dual line sinh2J=e2U. When J<U, we show that each of the two Ising configurations exhibits exponential decay of correlations while their product is ferromagnetically ordered.

Funding Statement

The first author is supported by the Swiss NSF grant P300P2_177848, Austrian Science Fund grant P3471 and partially supported by the European Research Council starting grant 678520 (LocalOrder). The second author is supported by Israel Science Foundation grants 861/15 and 1971/19, and by the European Research Council starting grant 678520 (LocalOrder).


The authors would like to thank Yinon Spinka for discussions concerning the FK–Ising representation of the six-vertex model and for spotting an error in an early draft of this paper — this allowed for a substantial extension of the results. Our conversations with Dmitry Chelkak, Hugo Duminil-Copin, Matan Harel, Marcin Lis, Ioan Manolescu, Aran Raoufi, and Gourab Ray were also very helpful.


Download Citation

Alexander Glazman. Ron Peled. "On the transition between the disordered and antiferroelectric phases of the 6-vertex model." Electron. J. Probab. 28 1 - 53, 2023.


Received: 17 April 2022; Accepted: 22 June 2023; Published: 2023
First available in Project Euclid: 11 July 2023

MathSciNet: MR4613855
zbMATH: 07721274
arXiv: 1909.03436
Digital Object Identifier: 10.1214/23-EJP980

Primary: 60K35 , 82B20

Keywords: Ashkin–Teller model , FKG inequality , Gibbs measures , height function , percolation , phase transition , Six-vertex model

Vol.28 • 2023
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