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Extreme gaps in the spectrum of Random Matrices

Random Matrix Theory and Its Applications I September 13, 2010 - September 17, 2010

September 14, 2010 (11:30 AM PDT - 12:10 PM PDT)
Speaker(s): Gerard Ben Arous (New York University, Courant Institute)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
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v0180

Abstract I will present a joint work with Paul Bourgade (Harvard) about the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices (CUE) and matrices from the Gaussian Unitary Ensemble. In particular, we show that the smallest gaps when rescaled by N-4/3, are Poissonian and we give the limiting distribution of the kth smallest gap. We also show that the largest gap, when normalized by √log N/N, converges in L^p to a constant for all p > 0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.
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