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Asymptotics of Tracy-Widom Distributions and the Total Integral of a Painlevé II Function

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Abstract

The Tracy-Widom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painlevé II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE Tracy-Widom distribution function, this gives an alternative proof of the recent work of Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom distribution functions are new.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Jinho Baik & Robert Buckingham

  2. Department of Mathematics, Seattle University, Seattle, WA, 98122, USA

    Jeffery DiFranco

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  1. Jinho Baik
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  2. Robert Buckingham
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  3. Jeffery DiFranco
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Correspondence to Robert Buckingham.

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Communicated by P. Sarnak

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Baik, J., Buckingham, R. & DiFranco, J. Asymptotics of Tracy-Widom Distributions and the Total Integral of a Painlevé II Function. Commun. Math. Phys. 280, 463–497 (2008). https://doi.org/10.1007/s00220-008-0433-5

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  • DOI: https://doi.org/10.1007/s00220-008-0433-5

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