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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 Painlev\'e II equation. Our results complement the ones in [Xu,Zhao,2011]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlev\'e II transcendents, and we also prove a new result on the poles of the Ablowitz-Segur solutions to the Painlev\'e II equation. We also highlight applications of our results in random matrix theory.

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          Author and article information

          Journal
          2015-07-07
          2016-03-26
          Article
          1507.01710
          64f1d7c7-63c1-49db-85ab-96d560fbd610

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          35 pages, 4 figures
          math-ph math.MP

          Mathematical physics,Mathematical & Computational physics
          Mathematical physics, Mathematical & Computational physics

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