Delocalization of the height function of the six-vertex model

We show that the height function of the six-vertex model, in the parameter range $\mathbf{a}=\mathbf{b}=1$ and $\mathbf{c}\ge 1$, is delocalized with logarithmic variance when $\mathbf{c}\le 2$. This complements the earlier proven localization for $\mathbf{c}>2$. Our proof relies on Russo–Seymour–Welsh type arguments, and on the local behaviour of the free energy of the cylindrical six-vertex model, as a function of the unbalance between the number of up and down arrows.