Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities

Abstract

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.

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Authors

Percy Deift

Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012

Alexander Its

Department of Mathematics
Indiana University -- Purdue University Indianapolis
402 N. Blackford St.
Indianapolis, IN 46292-3267

Igor Krasovsky

Department of Mathematical Sciences
Brunel University
Uxbridge UB8 3PH
United Kingdom

Current address:

Department of Mathematics
Imperial College
180 Queen's Gate
London SW7 2AZ
United Kingdom