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 $0<\Delta<1$ using convexity arguments and for large negative $\Delta$ using the Fixed Point Theory of appropriately defined contracting operator.