Exact Solutions to the Six-Vertex Model with Domain Wall Boundary Conditions and Uniform Asymptotics of Discrete Orthogonal Polynomials on an Infinite Lattice

Abstract

In this dissertation the partition function, Zn, for the six-vertex model with domain wall boundary conditions is solved in the thermodynamic limit in various regions of the phase diagram. In the ferroelectric phase region, we show that Zn=CGnFn2(1+O(e−n1−\ep)) for any \ep>0, and we give explicit formulae for the numbers C,G, and F. On the critical line separating the ferroelectric and disordered phase regions, we show that Zn=Cn1/4GnFn2(1+O(n−1/2)), and we give explicit formulae for the numbers G and F. In this phase region, the value of the constant C is unknown. In the antiferroelectric phase region, we show that Zn=C\th4(n\om)Fn2(1+O(n−1)), where \th4 is Jacobi's theta function, and explicit formulae are given for the numbers \om and F. The value of the constant C is unknown in this phase region. In each case, the proof is based on reformulating Zn as the eigenvalue partition function for a random matrix ensemble (as observed by Paul Zinn-Justin), and evaluation of large n asymptotics for a corresponding system of orthogonal polynomials. To deal with this problem in the antiferroelectric phase region, we consequently develop an asymptotic analysis, based on a Riemann-Hilbert approach, for orthogonal polynomials on an infinite regular lattice with respect to varying exponential weights. The general method and results of this analysis are given in Chapter 5 of this dissertation.