**CRM Monograph Series**

Volume: 32;
2014;
224 pp;
Hardcover

MSC: Primary 60;
Secondary 82

Print ISBN: 978-1-4704-0961-6

Product Code: CRMM/32

List Price: $98.00

Individual Member Price: $78.40

**Electronic ISBN: 978-1-4704-1442-9
Product Code: CRMM/32.E**

List Price: $98.00

Individual Member Price: $78.40

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#### Supplemental Materials

# Random Matrices and the Six-Vertex Model

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*Pavel Bleher; Karl Liechty*

A co-publication of the AMS and Centre de Recherches Mathématiques

This book provides a detailed description of the Riemann-Hilbert approach (RH approach) to the asymptotic analysis of both continuous and discrete orthogonal polynomials, and applications to random matrix models as well as to the six-vertex model. The RH approach was an important ingredient in the proofs of universality in unitary matrix models. This book gives an introduction to the unitary matrix models and discusses bulk and edge universality. The six-vertex model is an exactly solvable two-dimensional model in statistical physics, and thanks to the Izergin-Korepin formula for the model with domain wall boundary conditions, its partition function matches that of a unitary matrix model with nonpolynomial interaction. The authors introduce in this book the six-vertex model and include a proof of the Izergin-Korepin formula. Using the RH approach, they explicitly calculate the leading and subleading terms in the thermodynamic asymptotic behavior of the partition function of the six-vertex model with domain wall boundary conditions in all the three phases: disordered, ferroelectric, and antiferroelectric.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Graduate students and research mathematicians interested in random matrices and statistical mechanics.

#### Table of Contents

# Table of Contents

## Random Matrices and the Six-Vertex Model

- Cover Cover11
- Title page iii4
- Contents v6
- Introduction vii8
- Unitary matrix ensembles 112
- The Riemann-Hilbert problem for orthogonal polynomials 1930
- Discrete orthogonal polynomials on an infinite lattice 5566
- Introduction to the six-vertex model 8192
- The Izergin-Korepin formula 93104
- Disordered phase 109120
- Antiferroelectric phase 143154
- Ferroelectric phase 197208
- Between the phases 209220
- Bibliography 221232
- Back Cover Back Cover1237