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Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. \newl 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 {\it P. Deift}, {\it K. T. R. McLaughlin}, {\it T. Kriecherbauer}, {\it S. Venakides} and {\it 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 $\Sigma\subset\bbfC$ which are a finite union of (finite or infinite) smooth curves in $\bbfC$, 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 $v: \Sigma^0\to \text{GL}(n,\bbfC)$ which is smooth on $\Sigma^0$. The R-H problem consists in seeking an $n\times n$ matrix-valued function $m=m(z)$ which is: analytic in $\bbfC\setminus\Sigma$, $m_+(z)= m_-(z) v(z)$ $(\forall z\in \Sigma^0)$, $m(z)\to I$ as $z\to \infty$ where $m_{\pm}(z)$ denote the limits of $m(z')$ as $z'\to z$ 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.

47B80Random operators (linear)
47A56Functions whose values are linear operators
15B52Random matrices
30E25Boundary value problems, complex analysis
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
42C05General theory of orthogonal functions and polynomials
47B36Jacobi (tridiagonal) operators (matrices) and generalizations
60F99Limit theorems (probability)