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Dyson's Constant in the Asymptotics of the Fredholm Determinant of the Sine Kernel

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Abstract

We prove that the asymptotics of the Fredholm determinant of I−K α, where K α is the integral operator with the sine kernel on the interval [0, α], are given by

This formula was conjectured by Dyson. The proof for the first and second order asymptotics was given by Widom, and higher order asymptotics have also been determined. In this paper we identify the constant (or third order) term, which has been an outstanding problem for a long time.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Cruz, CA, 95064, USA

    Torsten Ehrhardt

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  1. Torsten Ehrhardt
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Correspondence to Torsten Ehrhardt.

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Communicated by J.L. Lebowitz

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Ehrhardt, T. Dyson's Constant in the Asymptotics of the Fredholm Determinant of the Sine Kernel. Commun. Math. Phys. 262, 317–341 (2006). https://doi.org/10.1007/s00220-005-1493-4

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  • DOI: https://doi.org/10.1007/s00220-005-1493-4

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