2008/03/31 by José A. Ramı́rez, Jose A. Ramirez, Brian Rider
Computer Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Bayesian Methods and Mixture Models #Random Matrices and Applications #math-ph #math.MP #math.PR
paper · pdf · doi:10.1007/s00220-008-0712-1
(More) typos corrected, added references
arxiv created 2008/10/08 · openalex publication_date 2009/01/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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.