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

作者：Liechty Karl^*

来源：Journal of Statistical Physics, 2012, 147(3): 582-622.

DOI：10.1007/s10955-012-0485-y

• Brownian excursions
• Airy process
• Random matrix theory
• Discrete orthogonal polynomials
• Painleve II

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 -> infinity, 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 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.

• 出版日期2012-5