Beta ensembles, Stochastic Airy spectrum, and a diffusion

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 − d2 dx2 + x+ 2√ β 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(β) distributions to all β> 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