UC Davis Mathematics

Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Exact solutions to the six-vertex model with domain wall boundary conditions

Mathematical Physics & Probability

Speaker: Karl kliechty, IUPUI
Location: 3106 MSB
Start time: Wed, Dec 1 2010, 4:10PM

The six vertex model is a two dimensional model of statistical mechanics realized on an $n \times n$ square lattice. In this talk, we consider specific fixed boundary conditions on this model known as domain wall boundary conditions (DWBC). With these boundary conditions, states of the six-vertex model are in bijection with the set of $n \times n$ alternating sign matrices. A very nice determinantal formula for the partition function of the six-vertex model with DWBC was derived by Izergin and Korepin in the 1980's. About 10 years ago, Paul Zinn-Justin noticed that the Izergin-Korepin formula can be written in the form of the partition function for a random matrix-type ensemble, making it amenable to asymptotic analysis. In this talk, I will discuss the Izergin-Korepin formula, the random matrix interpretation, and the asymptotic analysis, which is based on the Riemann-Hilbert approach.