Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. (Q2748505)

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 \textit{P. Deift}, \textit{K. T. R. McLaughlin}, \textit{T. Kriecherbauer}, \textit{S. Venakides} and \textit{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 \(\Sigma\subset\mathbb{C}\) which are a finite union of (finite or infinite) smooth curves in \(\mathbb{C}\), i.e. the curves intersect at most at a finite number of points and all intersections are transversal.NEWLINENEWLINENEWLINEChapter 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 \(v: \Sigma^0\to \text{GL}(n,\mathbb{C})\) 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 \(\mathbb{C}\setminus\Sigma\), \(m_+(z)= m_-(z) v(z)\) \((\forall z\in \Sigma^0)\), \(m(z)\to I\) 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 \(\Sigma\).NEWLINENEWLINENEWLINEIn 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.NEWLINENEWLINENEWLINEChapter 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.NEWLINENEWLINENEWLINEThe 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.