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A diffusion description of the random matrix hard edge

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With J. Ramirez and B. Virag we recently proved that the limiting soft edge eigenvalues of the general beta ensembles have laws shared by the spectral points of a certain random Schroedinger operator. After recalling this fact I’ll prove there is a similar picture at the random matrix hard edge. That is, the small eigenvalues of sample covariance ensembles are described in terms of a (random) differential operator in the large dimensional limit. Via a Riccati transformation, there is a second description though the hitting distributions of a simple diffusion. The latter picture allows a proof of the anticipated transition between the hard and soft edge laws.

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