Delocalization of the height function of the six-vertex model

Abstract

We show that the height function of the six-vertex model, in the parameter range \mathbf{a}=\mathbf{b}=1 and \mathbf{c}\ge1 , 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.