Beta ensembles, stochastic Airy spectrum, and a diffusion

Abstract

We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schroedinger operator -d^2/dx^2 + x + (2/beta^{1/2}) b_x' restricted to the positive half-line, where b_x' is white noise. In doing so we extend the definition of the Tracy-Widom(beta) distributions to all beta>0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.

Revised content, new results. In particular, Theorems 1.3 and 5.1 are new

References in corpus (6)

• Continuum limits of random matrices and the Brownian carousel
• From Random Matrices to Stochastic Operators
• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• Diffusion at the random matrix hard edge
• Hermite and Laguerre $\beta$-ensembles: asymptotic corrections to the eigenvalue density
• Large gaps between random eigenvalues

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