We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure log 2 1−x dx on (−1, 1). The asymptotic formula confirms a special case of a conjecture by A. Magnus and extends earlier results by T. O. Conway and one of the authors. The proof relies on the Riemann-Hilbert method. The main difficulty in applying the method to the problem at hand is the lack of an appropriate local parametrix near the logarithmic singularity at x = +1.

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The authors consider the length, l N , of the length of the longest increasing subsequence of a random permutation of N numbers. The main result in this paper is a proof that the distribution function for l N , suitably centered and scaled, converges to the Tracy-Widom distribution [TW1] of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest decent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 [DZ1] in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel [Ge] for the Poissonization of the distribution function of l N .

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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 q k x k , q 2m > 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.

We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general nondegenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.