On the thermodynamic limit of the 6-vertex model

We give a rigorous treatment to the thermodynamic limit of the 6-vertex model. We prove that the unique solution of the Bethe-Ansatz equation exists and the distribution of the roots converges to a continuum measure. We solve this problem for $0<\mathrm{\Delta }<1$ using convexity arguments and for large negative $\mathrm{\Delta }$ using the Fixed Point Theory of appropriately defined contracting operator.