Exact solutions to the six-vertex model with domain wall boundary conditions and uniform asymptotics of discrete orthogonal polynomials on an infinite lattice.

摘要

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 = CG n Fn2 (1 + O( e-n1-3 )) for any epsilon > 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 = Cn1/4 GnFn2 (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 Z n = Ctheta4(no ) Fn2 (1 + O(n-1)), where theta4 is Jacobi's theta function, and explicit formulae are given for the numbers o 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.