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To obtain Russo-Seymour-Welsh estimates for the height function of thesix-vertex model under sloped boundary conditions, which can be leveraged todemonstrate that the height function logarithmically delocalizes under abroader class of boundary conditions, we formulate crossing probabilityestimates in strips of the square lattice and the cylinder, for parameterssatisfying $a\equiv b$, $c \in [1,2]$, and $\mathrm{max} \{ a , b \} \leq c$,in which each of the first two conditions respectively relate to invarianceunder vertical and diagonal reflections enforced through the symmetry $\sigma\xi \geq -\xi$ for domains in strips of the square lattice, and satisfaction ofFKG, for the height function and for its absolute value. To determine whetherarguments for estimating crossing probabilities of the height function for flatboundary conditions from a recent work due to Duminil-Copin, Karila, Manolescu,and Oulamara remain applicable for sloped boundary conditions, from the set ofpossible slopes given by the rational points from $[-1,1] \times [-1,1]$, weanalyze sloped Gibbs states, which do not have infinitely many disjointlyoriented circuits. In comparison to Russo-Seymour-Welsh arguments for flatboundary conditions, arguments for sloped boundary conditions presentadditional complications for both planar and cylindrical settings, in whichcrossing events that are considered in the strip, and then extended to theannulus and cylinder, must be maintained across rectangles of large aspectratio, in spite of the fact that some proportion of faces within the stripfreeze with positive probability. Once considerations have been raised forestimating crossing probabilities in the strip, and in the cylinder, whichrelate to the fraction of points about which cylindrical domains are situatedthat are deemed to satisfy the \textit{good} property, we investigate othermodels of interest.

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