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Nonintersecting Brownian Motions on the Half-Line and Discrete Gaussian Orthogonal Polynomials

Authors
  • Liechty, Karl1
  • 1 University of Michigan, Department of Mathematics, 530 Church St., Ann Arbor, MI, 48109, USA , Ann Arbor (United States)
Type
Published Article
Journal
Journal of Statistical Physics
Publisher
Springer-Verlag
Publication Date
May 04, 2012
Volume
147
Issue
3
Pages
582–622
Identifiers
DOI: 10.1007/s10955-012-0485-y
Source
Springer Nature
Keywords
License
Yellow

Abstract

We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the Airy2 process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.

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