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

TL;DR

This paper analyzes the maximal height distribution of nonintersecting Brownian motions on a half-line, demonstrating its convergence to the Tracy-Widom distribution through Riemann-Hilbert analysis of discrete orthogonal polynomials.

Contribution

It introduces a Riemann-Hilbert approach to study the asymptotics of discrete orthogonal polynomials in the double scaling limit, linking Brownian motion maxima to random matrix theory.

Findings

Maximal height converges to Tracy-Widom distribution.

Asymptotics of free energy and kernel computed in critical scaling.

Results are dual to eigenvalue density vanishing in random matrix models.

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 $\textrm{Airy}_2$ 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.