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 Schrödinger operator $-\frac {d^2}{dx^2} + x + \frac {2}{\sqrt {\beta }} b_x^{\prime }$ restricted to the positive half-line, where $b_x^{\prime }$ 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 converges to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.