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 MathAcmatiques.The theory of random matrices has proven to have a wide reach into many areas of mathematics, physics, and statistics, and there are many excellent books on the topic. ... See also the reviews [34] by Di Francesco, Ginsparg, and Zinn-Justin; the ones in the MSRI volume [12], edited by Bleher ... In particular, the Riemanna Hilbert method allows for an asymptotic analysis of a wide class of orthogonal polynomials, ... In that paper certain recursions for the partition function were derived.

Title | : | Random Matrices and the Six-Vertex Model |

Author | : | Pavel Bleher, Karl Liechty |

Publisher | : | American Mathematical Soc. - 2013-12-04 |

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