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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 Σ which are a finite union of (finite or infinite) smooth curves in , 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 Σ 0 =Σ{points of self-intersection of Σ}. Suppose in addition that there exists a map v:Σ 0 GL(n,) which is smooth on Σ 0 . The R-H problem consists in seeking an n×n matrix-valued function m=m(z) which is: analytic in Σ, m + (z)=m - (z)v(z) (zΣ 0 ), m(z)I as z where m ± (z) denote the limits of m(z ' ) as z ' z from the positive (resp. negative) side of Σ.

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:
47B80Random operators (linear)
47A56Functions whose values are linear operators
15A52Random matrices (MSC2000)
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)