On the transition between the disordered and antiferroelectric phases of the 6-vertex model
The symmetric six-vertex model with parameters $a,b,c>0$ is expected to exhibit different behavior in the regimes $a+b<c$ (antiferroelectric), $|a-b|<c\leq a+b$ (disordered) and $|a-b|>c$ (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 ($\mathbb{T}$-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 $\mathbb{Z}^2$ at its self-dual line $\sinh 2J = e^{-2U}$. When $J<U$, we show that each of the two Ising configurations exhibits exponential decay of correlations while their product is ferromagnetically ordered.