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On the asymptotics of a Toeplitz determinant with singularities. (English) Zbl 1326.35218

Deift, Percy (ed.) et al., Random matrix theory, interacting particle systems and integrable systems. New York, NY: Cambridge University Press (ISBN 978-1-107-07992-2/hbk). Mathematical Sciences Research Institute Publications 65, 93-146 (2014).
Summary: We provide an alternative proof of the classical single-term asymptotics for Toeplitz determinants whose symbols possess Fisher-Hartwig singularities. We also relax the smoothness conditions on the regular part of the symbols and obtain an estimate for the error term in the asymptotics. Our proof is based on the Riemann-Hilbert analysis of the related systems of orthogonal polynomials and on differential identities for Toeplitz determinants. The result discussed in this paper is crucial for the proof of the asymptotics in the general case of Fisher-Hartwig’s singularities and extensions to Hankel and Toeplitz+Hankel determinants.
For the entire collection see [Zbl 1318.81009].

MSC:

35Q15 Riemann-Hilbert problems in context of PDEs
15B05 Toeplitz, Cauchy, and related matrices
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C52 Orthogonal polynomials and functions associated with root systems
60H25 Random operators and equations (aspects of stochastic analysis)
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
35B40 Asymptotic behavior of solutions to PDEs
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