182 research outputs found
Normality of Circular -ensemble
We will prove the Berry-Esseen theorem for the number counting function of
the circular -ensemble (CE), which will imply the central limit
theorem for the number of points in arcs. We will prove the main result by
estimating the characteristic functions of the Pr\"ufer phases and the number
counting function, which will imply the the uniform upper and lower bounds of
their variance. We also show that the similar results hold for the Sine
process. As a direct application of the uniform variance bound, we can prove
the normality of the linear statistics when the test function for some
Large gaps of CUE and GUE
In this article, we study the largest gaps of the classical random matrices
of CUE and GUE, and we will derive the rescaling limit of the -th largest
gap, which is given by the Gumbel distribution
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