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UCCS MATHEMATICS COLLOQUIUM

A Diffusion View of the Random Matrix Edge

Abstract: We prove that the general Tracy-Widom laws at the soft edge of RMT's standard beta ensembles are shared by those at the bottom of the spectrum of a certain family of random Schroedinger operators, settling a conjecture of Edelman and Sutton. By the classical Riccati map, we provide a second description of these laws in terms of the explosion probability of a simple one-dimensional diffusion process. A similar picture holds at the limit of the hard edge.