96-537 Alexander Soshnikov

Global Level Spacings Distribution for Large Random Matrices : Gaussian Fluctuations (340K, ps) Oct 29, 96

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Abstract. We study the level-spacings distribution for eigenvalues of large $ \ N \times N \ $ matrices from the Classical Compact Groups in the scaling limit when the mean distance between nearest eigenvalues equals 1.\\ Defining by $\ \eta_N(s) \ $ the number of nearest neighbors spacings, greater tnan $ \ s>0 \ $ (smaller than $ \ s>0 \ $ ) we prove functional limit theorem for the process \\ $ (\eta_N(s)-E \eta_N(s))/N^{1/2} $, giving weak convergence of this distribution to some Gaussian random process on $ \ \ \ [0, \infty ) \ \ $.\\ The limiting Gaussian random process is universal for all Classical Compact Groups. It is H\"older continuous with any exponent less than $ \ \ 1/2 \ \ .$ Numerical results suggest it not to be a standard Brownian bridge.\\ Our methods can be also applied to study n-level spacings distribution.

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