Riemann-Hilbert Approach to the Six-Vertex Model

Abstract

The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $n$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an $n\times n$ Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large $n$ asymptotics of the six-vertex model with DWBC. The solution is based on the Riemann-Hilbert approach. In this paper we review asymptotic results obtained in different regions of the phase diagram.

arXiv comment15 pages, 4 figures, Submitted to the MSRI volume on "Random Matrix Theory, Interacting Particle Systems and Integrable Systems". arXiv admin note: text overlap with arXiv:math-ph/0510033