  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.
