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Supremum of the Airy2 Process Minus a Parabola on a Half Line

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Abstract

Let \(\mathcal {A}_{2}(t)\) be the Airy2 process. We show that the random variable

$$\sup_{t\leq\alpha} \bigl\{\mathcal {A}_2(t)-t^2 \bigr\}+\min\{0,\alpha \}^2 $$

has the same distribution as the one-point marginal of the Airy2→1 process at time α. These marginals form a family of distributions crossing over from the GUE Tracy-Widom distribution F GUE(x) for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE Tracy-Widom distribution F GOE(41/3 x) for the Gaussian Orthogonal Ensemble. Furthermore, we show that for every α the distribution has the same right tail decay \(e^{-\frac{4}{3} x^{3/2} }\).

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Notes

  1. This corresponds to a minor correction of the formula appearing in [7], where the exponential prefactor appears in front of L 0 instead of L 1+L 2.

  2. This is due to the known asymptotics \(\log (1-F_{\mathrm{GOE}}(m) )\sim-\frac{2}{3}m^{3/2}\) and \(\log (1-F_{\mathrm{GUE}}(m) )\sim-\frac{4}{3}m^{3/2}\), which follow from the formulas for these distributions in terms of the Painlevé II function [30, 31].

  3. The Baker-Campbell-Hausdorff formula can be found in most introductory books on Lie groups and algebras. A general version can be found in [15]. However, in this very simple context, it is more readily computed by hand.

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Acknowledgements

J.Q. and D.R. were supported by the Natural Science and Engineering Research Council of Canada, and D.R. was supported by a Fields-Ontario Postdoctoral Fellowship and by Fondecyt Grant 1120309. Part of this work was done during the Fields Institute program “Dynamics and Transport in Disordered Systems” and the authors would like to thank the Fields Institute for its hospitality.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada, M5S 2E4

    Jeremy Quastel & Daniel Remenik

  2. Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile

    Daniel Remenik

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  1. Jeremy Quastel
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Correspondence to Jeremy Quastel.

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Quastel, J., Remenik, D. Supremum of the Airy2 Process Minus a Parabola on a Half Line. J Stat Phys 150, 442–456 (2013). https://doi.org/10.1007/s10955-012-0633-4

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  • DOI: https://doi.org/10.1007/s10955-012-0633-4

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