From logarithmic delocalization of the six-vertex height function under sloped boundary conditions to weakened crossing probability estimates for the Ashkin-Teller, generalized random-cluster, and $(q_{\sigma},q_{\tau})$-cubic models

TL;DR

This paper develops crossing probability estimates for the six-vertex model's height function under sloped boundary conditions, demonstrating logarithmic delocalization and extending RSW techniques to more general boundary settings.

Contribution

It introduces new crossing probability estimates for the six-vertex model with sloped boundary conditions, broadening the applicability of RSW methods beyond flat boundaries.

Findings

Height function logarithmically delocalizes under broader boundary conditions.

Crossing probabilities can be estimated in strips and cylinders with sloped boundaries.

Arguments for flat boundary conditions are extended to more complex boundary geometries.

Abstract

To obtain Russo-Seymour-Welsh estimates for the height function of the six-vertex model under sloped boundary conditions, which can be leveraged to demonstrate that the height function logarithmically delocalizes under a broader class of boundary conditions, we formulate crossing probability estimates in strips of the square lattice and the cylinder, for parameters satisfying $a\equiv b$, $c \in [1,2]$, and $\mathrm{max} \{ a , b \} \leq c$, in which each of the first two conditions respectively relate to invariance under vertical and diagonal reflections enforced through the symmetry $\sigma \xi \geq -\xi$ for domains in strips of the square lattice, and satisfaction of FKG, for the height function and for its absolute value. To determine whether arguments for estimating crossing probabilities of the height function for flat boundary conditions from a recent work due to Duminil-Copin, Karila, Manolescu, and Oulamara remain applicable for sloped boundary conditions, from the set of possible slopes given by the interior of the set of rational points from $[-1,1] \times [-1,1]$, we analyze sloped Gibbs states, which do not have infinitely many disjointly oriented circuits. In comparison to Russo-Seymour-Welsh arguments for flat boundary conditions, arguments for sloped boundary conditions present additional complications for both planar and cylindrical settings, in which crossing events that are considered in the strip, and then extended to the annulus and cylinder, must be maintained across rectangles of large aspect ratio, in spite of the fact that some proportion of faces within the strip freeze with positive probability.