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

Reprint of the 1998 original.

(English) Zbl 0997.47033
Courant Lecture Notes in Mathematics. 3. New York, NY: Courant Institute of Mathematical Sciences. Providence, RI: American Mathematical Society (AMS). ix, 261 p. \$ 24.00 (2000).

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 ${\Sigma }\subset ℂ$ which are a finite union of (finite or infinite) smooth curves in $ℂ$, 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 ${{\Sigma }}^{0}={\Sigma }\setminus \left\{$points of self-intersection of ${\Sigma }\right\}$. Suppose in addition that there exists a map $v:{{\Sigma }}^{0}\to \text{GL}\left(n,ℂ\right)$ which is smooth on ${{\Sigma }}^{0}$. The R-H problem consists in seeking an $n×n$ matrix-valued function $m=m\left(z\right)$ which is: analytic in $ℂ\setminus {\Sigma }$, ${m}_{+}\left(z\right)={m}_{-}\left(z\right)v\left(z\right)$ $\left(\forall z\in {{\Sigma }}^{0}\right)$, $m\left(z\right)\to I$ as $z\to \infty$ where ${m}_{±}\left(z\right)$ 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)