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

In this dissertation the partition function, Z_n, 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 Z_n=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 Z_n=Cn^1/4G^\sqrt{n}Fⁿ²(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=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 Z_n 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.

Author dc:creator

• Liechty, Karl Edmund

Advisor dc:contributor.advisor

• Bleher, Pavel, 1947-

• Statistical Mechanics, Random Matrices, Orthogonal Polynomials, Asymptotics, Riemann-Hilbert Problems

Language dc:language.iso

en_US

Identifier URI

http://dx.doi.org/10.7912/C2/2390

OAI identifier oai:identifier

oai:scholarworks.indianapolis.iu.edu:1805/2482