Orthogonal polynomials and random matrices: a Riemann-Hilbert approach.

Reprint of the 1998 original.

*(English)*Zbl 0997.47033

Apart of certain additional preparatory material, this book is a pedagogic illustration of the general methods and results of the author with his collaborators in 1997-1999, in a special case (see chapters 7-8) in which the technical difficulties are at a minimum. The above general methods/results are presented in the long papers of *P. Deift*, *K. T. R. McLaughlin*, *T. Kriecherbauer*, *S. Venakides* and *X. Zhou* [Int. Math. Res. Not. 1997, No. 16, 759-782 (1997; Zbl 0897.42015), Comm. Pure Appl. Math. 52, No. 11, 1335-1425 (1999; Zbl 0944.42013), ibid. 52, No. 12, 1491-1552 (1999; Zbl 1026.42024)]. The above particular case means in particular that the author considers in this book only contours ${\Sigma}\subset \u2102$ which are a finite union of (finite or infinite) smooth curves in $\u2102$, i.e. the curves intersect at most at a finite number of points and all intersections are transversal.

Chapter 1 devotes the definition of the $n$-dimensional Riemann-Hilbert (R-H) problem and some applied examples. Let ${{\Sigma}}^{0}={\Sigma}\setminus \{$points of self-intersection of ${\Sigma}\}$. Suppose in addition that there exists a map $v:{{\Sigma}}^{0}\to \text{GL}(n,\u2102)$ which is smooth on ${{\Sigma}}^{0}$. The R-H problem consists in seeking an $n\times n$ matrix-valued function $m=m\left(z\right)$ which is: analytic in $\u2102\setminus {\Sigma}$, ${m}_{+}\left(z\right)={m}_{-}\left(z\right)v\left(z\right)$ $(\forall z\in {{\Sigma}}^{0})$, $m\left(z\right)\to I$ as $z\to \infty $ where ${m}_{\pm}\left(z\right)$ denote the limits of $m\left({z}^{\text{'}}\right)$ as ${z}^{\text{'}}\to z$ from the positive (resp. negative) side of ${\Sigma}$.

In Chapters 2 and 3 the author presents with proofs a number of basic and well-known facts from the classical theory of orthogonal polynomials and Jacobi matrices/operators in the way of relating this theory to the theory of R-H problems.

Chapter 4 devotes to some aspects of the beautiful relationship between continuous fractions and orthogonal polynomials. Chapter 5 devotes basic concepts of random matrix theory. Chapter 6 devotes the principal properties of equilibrium measures.

The crucial Chapters 7-8 devote the above results for asymptotics of orthogonal polynomials and their applications to universality questions in random matrix theory via the corresponding R-H problems.

##### MSC:

47B80 | Random operators (linear) |

47A56 | Functions whose values are linear operators |

15A52 | Random matrices (MSC2000) |

30E25 | Boundary value problems, complex analysis |

33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |

37K10 | Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies |

42C05 | General theory of orthogonal functions and polynomials |

47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |

60F99 | Limit theorems (probability) |