The characteristic polynomial of a random permutation matrix at different points

Dang, Kim; Zeindler, Dirk

doi:10.1016/j.spa.2013.08.003

Cited by 8 publications

(9 citation statements)

References 26 publications

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“…Also, for m = 1 and for random permutation matrices without modification, the result simply derives from Theorem 1.5 in [3]. Furthermore, the third item can be deduced from Proposition 1.2 in [7] considering the imaginary part of the logarithm of the characteristic polynomial, for the specific case where the family (1, α 1 , · · · , α m , β 1 , · · · , β m ) is linearly independent over Z. Theorem 2. Let I 1 , · · · , I m be a finite number of fixed arcs of the form…”

Section: Notations and Main Resultsmentioning

confidence: 99%

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

Bahier

2017

J Theor Probab

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We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens' distribution of a given parameter 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 behaviour of the largest and smallest spacing between two distinct consecutive eigenvalues.

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Section: Notations and Main Resultsmentioning

confidence: 99%

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

Bahier

2017

J Theor Probab

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“…The following result has been first established by Wieand [13] [12] in the particular case θ = 1, then by Ben Arous and Dang [5] for permutation matrices in the general case θ > 0, and can be deduced under stronger assumptions from a result of Dang and Zeindler [6] on the logarithm of the characteristic polynomial of permutation matrices. Proposition 1.…”

Section: Macroscopic Scalementioning

confidence: 95%

On a limiting point process related to modified permutation matrices

Bahier¹

2020

ALEA

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We 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. For each of these two ensembles of matrices, rescaling properly the eigenangles provides a limiting point process as the size of the matrices goes to infinity. If J is an interval of R, we show that, as the length of J tends to infinity, the number of points lying in J of the limiting point process related to modified permutation matrices is asymptotically normal. Moreover, for permutation matrices without modification, if a and a + b denote the endpoints of J, we still have an asymptotic normality for the number of points lying in J, in the two following cases: [a fixed and b → ∞] and [a, b → ∞ with b proportional to a].See [4] for a proof of the two last asymptotic equalities. Mesoscopic scaleIn [4], the author of the present paper establishes the following result: Proposition 2. Assume I to be depending on n, of the form I = I n := e 2iπα , e 2iπ(α+δn) , where α ∈ [0, 1) and (δ n ) is a sequence of positive real numbers satisfying δ n −→ n→∞ 0 nδ n −→ n→∞ +∞.

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“…Zeindler [20] [21] generalizes this result for permutation matrices under Ewens measures, considering more general class functions than the characteristic polynomial, the so-called multiplicative class functions. Dehaye and Zeindler [6], and Dang and Zeindler [5] extend the study to some Weyl groups, and some wreath products involving the symmetric group.…”

Section: Convergence Of Characteristic Polynomialsmentioning

confidence: 99%

“…Based on the fact that all eigenvalues of permutation matrices are roots of unity, then, for every irrational number α, z = e 2iπα is almost surely not a zero of Z n for all n. Let α be an irrational number between 0 and 1. It is natural to shift the random process of eigenangles by 2πα, and consider the function ξ n,α defined by (5).…”

Section: Quotient Of Characteristic Polynomials Related To Permutatiomentioning

confidence: 99%

Characteristic polynomials of modified permutation matrices at microscopic scale

Bahier

2019

Stochastic Processes and their Applications

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We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation group. We also look at some modifications of permutation matrices where the entries equal to one are replaced by i.i.d uniform variables on the unit circle. Once appropriately normalized and scaled, we show that the characteristic polynomial converges in distribution on every compact subset of C to an explicit limiting entire function, when the size of the matrices goes to infinity. Our findings can be related to results by Chhaibi, Najnudel and Nikeghbali on the limiting characteristic polynomial of the Circular Unitary Ensemble [4].

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The characteristic polynomial of a random permutation matrix at different points

Cited by 8 publications

References 26 publications

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

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

On a limiting point process related to modified permutation matrices

Characteristic polynomials of modified permutation matrices at microscopic scale

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