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Hankel Determinant and Orthogonal Polynomials for a Gaussian Weight with a Discontinuity at the Edge

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Abstract

We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz–Segur family of solutions to the Painlevé II equation. Our results complement the ones in [33]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlevé II transcendents, and we also prove a new result on the poles of the Ablowitz–Segur solutions to the Painlevé II equation. We also highlight applications of our results in random matrix theory.

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Authors and Affiliations

  1. Department of Higher Mathematics and Mathematical Physics, Saint-Petersburg State University, Saint-Petersburg, Russia

    A. Bogatskiy

  2. Department of Physics, Saint-Petersburg State University, Saint-Petersburg, Russia

    A. Bogatskiy

  3. Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du Cyclotron 2, B-1348, Louvain-La-Neuve, Belgium

    T. Claeys

  4. Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, IN, 46202-3216, USA

    A. Its

Authors
  1. A. Bogatskiy
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  2. T. Claeys
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  3. A. Its
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Correspondence to A. Bogatskiy.

Additional information

Communicated by P. Deift

At the time of publication, A. Bogatskiy is a student at the University of Chicago.

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Bogatskiy, A., Claeys, T. & Its, A. Hankel Determinant and Orthogonal Polynomials for a Gaussian Weight with a Discontinuity at the Edge. Commun. Math. Phys. 347, 127–162 (2016). https://doi.org/10.1007/s00220-016-2691-y

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