Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase

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 × 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 in the disordered phase. The solution is based on the Riemann–Hilbert approach and the Deift–Zhou nonlinear steepest descent method. As was noticed by Kuperberg, the problem of enumeration of alternating sign matrices (the ASM problem) is a special case of the six-vertex model. We compare the obtained exact solution of the six-vertex model with known exact results for the 1, 2, and 3 enumerations of ASMs, and also with the exact solution on the so-called free fermion line. We prove the conjecture of Zinn–Justin that the partition function of the six-vertex model with DWBC has the asymptotics, as N → ∞, and we find the exact value of the exponent κ.The first author is supported in part by the National Science Foundation (NSF) Grant DMS-0354962.