On the Six-Vertex Model’s Free Energy

Abstract

AbstractIn this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime$$\Delta <1$$Δ<1. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when$$a=b=1$$a=b=1and$$c\ge 1$$c≥1, and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.