On the Number of Eigenvalues of Modified Permutation Matrices in Mesoscopic Intervals

Valentin Bahier ()
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Valentin Bahier: Institut de Mathématiques de Toulouse

Journal of Theoretical Probability, 2019, vol. 32, issue 2, 974-1022

Abstract: Abstract We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens’ distribution of a given parameter $$\theta >0$$ θ > 0 and its modification where entries equal to 1 in the matrices are replaced by independent random variables uniformly distributed on the unit circle. For the elements of each ensemble, we focus on the random numbers of eigenvalues lying in some specified arcs of the unit circle. We show that for a finite number of fixed arcs, the fluctuation of the numbers of eigenvalues belonging to them is asymptotically Gaussian. Moreover, for a single arc, we extend this result to the case where the length goes to zero sufficiently slowly when the size of the matrix goes to infinity. Finally, we investigate the behavior of the largest and smallest spacings between two distinct consecutive eigenvalues.

Keywords: Random matrices; Permutations; Distribution of eigenvalues; Wreath product; Central limit theorem; 60B20; 60F05; 20B30 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-017-0798-5

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