In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed Orthogonal polynomials on the unit circle are not discussed

Orthogonal Polynomials

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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An Introduction to Random Matrices

Let SN be the group of permutations of 1,2,..., N. If 7r E SN, we say that 7(i1),... , 7F(ik) is an increasing subsequence in 7r if il < i2 < ... < ik and 7r(ii) < 7r(i2) < ...< 7r(ik). Let 1N(r) be the length of the longest increasing subsequence. For example, if N = 5 and 7r is the permutation 5 1 3 2 4 (in one-line notation: thus 7r(1) = 5, 7r(2) = 1, ... ), then the longest increasing subsequences are 1 2 4 and 1 3 4, and N() = 3. Equip SN with uniform distribution,

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On the distribution of the length of the longest increasing subsequence of random permutations

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

Shape Fluctuations and Random Matrices

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel-Rotach-type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]).



The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμV for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx= e Q(x) dx on the real line, where Q(x)=∑ 2m k=0 qkx k , q2m> 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. c 1999 John Wiley & Sons, Inc.

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Strong asymptotics of orthogonal polynomials with respect to exponential weights

New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field

This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrodinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Holder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.

Kenneth D.T-R McLaughlin

Papers

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

Strong asymptotics of orthogonal polynomials with respect to exponential weights

New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field

Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation

Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration