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 < ∆ < 1 using convexity arguments and for large negative ∆ using the Fixed Point Theory of appropriately defined contracting operator. 1 The 6-vertex model and formulation of the problem The 6-vertex model is an exactly soluble model of classical statistical mechanics introduced and solved in various special cases by Lieb [1, 2, 3]. A solution of the most general case was obtained by Sutherland [4]. A clear description of this model and various other soluble models can be found in Baxter's book [5]. However, as Baxter remarks, an exact solution is not the same as a rigorous solution. In fact, already in his first article on the ice model [1], Lieb initiated the rigorous analysis of the model. A more extensive analysis was made by Lieb and Wu [6]. An important technical question was left unresolved, however. This concerns the convergence of the distribution of (quasi-) wavenumbers to a continuum measure in the thermodynamic limit. (Another technical problem, i.e. the independence of the free energy on the boundary conditions, was resolved by Brascamp et al. [7].) A similar 1