Indiana University-Purdue University Indianapolis (IUPUI) / 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 $Z_n=CG^nF^{n^2}(1+O(e^{-n^{1-\ep}}))$ 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 $Z_n=Cn^{1/4}G^{\sqrt{n}}F^{n^2}(1+O(n^{-1/2}))$, 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 $Z_n=C\th_4(n\om)F^{n^2}(1+O(n^{-1}))$, where $\th_4$ 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.
| Identifer | oai:union.ndltd.org:IUPUI/oai:scholarworks.iupui.edu:1805/2482 |
| Date | 09 March 2011 |
| Creators | Liechty, Karl Edmund |
| Contributors | Bleher, Pavel, 1947-, Its, Alexander R., Lempert, Lazlo, Kitchens, Bruce, 1953- |
| Source Sets | Indiana University-Purdue University Indianapolis |
| Language | en_US |
| Detected Language | English |
| Type | Thesis |
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