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Level Spacings Distribution for Large Random Matrices: Gaussian Fluctuations

Alexander Soshnikov
Annals of Mathematics
Second Series, Vol. 148, No. 2 (Sep., 1998), pp. 573-617
Published by: Annals of Mathematics
DOI: 10.2307/121004
Stable URL: http://www.jstor.org/stable/121004
Page Count: 45
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Level Spacings Distribution for Large Random Matrices: Gaussian Fluctuations
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Abstract

We study the level-spacings distribution for eigenvalues of large N × N matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1. Defining by η N(s) the number of nearest neighbors spacings greater than s > 0 (smaller than s > 0) we prove functional limit theorem for the process (η N(s) - Eη N(s))/N1/2, giving weak convergence of this distribution to some Gaussian random process on [0, ∞ ). The limiting Gaussian random process is universal for all classical compact groups. It is Hölder continuous with any exponent less than 1/2. Similar results can be obtained for the n-level-spacings distribution.

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