  We show that the limiting minimal eigenvalue distributions for a natural
generalization of Gaussian sample-covariance structures (the "beta ensembles")
are described by the spectrum of a random diffusion generator. By a Riccati
transformation, we obtain a second diffusion description of the limiting
eigenvalues in terms of hitting laws. This picture pertains to the so-called
hard edge of random matrix theory and sits in complement to the recent work of
the authors and B. Virag on the general beta random matrix soft edge. In fact,
the diffusion descriptions found on both sides are used here to prove there
exists a transition between the soft and hard edge laws at all values of beta.
