Diffusion at the random matrix hard edge

We show that the limiting minimal eigenvalue distributions for a natural general-ization of Gaussian sample-covariance structures (beta ensembles) are described by the spectrum of a random diffusion generator. This generator may be mapped onto the “Stochastic Bessel Operator,” introduced and studied by A. Edelman and B. Sutton in [6] where the corresponding convergence was first conjectured. Here, by a Riccati transformation, we also obtain a second diffusion description of the limiting eigenvalues in terms of hitting laws. All this pertains to the so-called hard edge of random matrix theory and sits in complement to the recent work [15] of the authors and B. Virág on the general beta random matrix soft edge. In fact, the diffusion descriptions found on both sides are used below to prove there exists a transition between the soft and hard edge laws at all values of beta