Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials

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 $N\to \infty$, 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 ${\text{Airy}}_2$ 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.

39 pages, 4 figures; The title has been changed from "The limiting distribution of the maximal height of nonintersecting Brownian excursions and discrete Gaussian orthogonal polynomials." This is a reflection of the fact that the analysis has been adapted to include nonintersecting Brownian motions with either reflecting of absorbing boundaries at zero. To appear in J. Stat. Phys

References in corpus (7)

• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Exact distribution of the maximal height of p vicious walkers
• Nonintersecting Brownian excursions
• Distribution of the time at which N vicious walkers reach their maximal height
• Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices
• Maximum distributions of bridges of noncolliding Brownian paths

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