On the Number of Eigenvalues of Modified Permutation Matrices in Mesoscopic Intervals
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In this paper, the authors considered the ensemble of permutation matrices following Ewens' distribution of a given parameter and its modification where entries equal to 1 in the matrices are replaced by independent random variables uniformly distributed on the unit circle.Abstract:
We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens’ distribution of a given parameter $$\theta >0$$
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.read more
Citations
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Eigenvalue Fluctuations of Symmetric Group Permutation Representations on k-tuples and k-subsets
TL;DR: In this paper, it was shown that the normalized count of the number of eigenangles in a fixed interval of a permutation representation evaluated at a random element of the symmetric group σ ∈ \mathfrak{S}_n$ converges weakly to a compactly supported distribution.
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On a limiting point process related to modified permutation matrices
TL;DR: In this article, the authors consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle.
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On a limiting point process related to modified permutation matrices.
TL;DR: In this article, the authors consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle.
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On smooth mesoscopic linear statistics of the eigenvalues of random permutation matrices.
Valentin Bahier,Joseph Najnudel +1 more
TL;DR: In this article, the authors studied the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group.
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On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random Permutation Matrices
Valentin Bahier,Joseph Najnudel +1 more
TL;DR: In this article, the authors studied the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group.
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