Permutation Matrices and the Moments of their Characteristics Polynomials

Zeindler, Dirk

doi:10.1214/ejp.v15-781

Cited by 19 publications

(26 citation statements)

References 13 publications

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“…the case of a onecycle permutation, and observing that the characteristic polynomial factors when the permutation matrix decomposes into blocks. More explicit details can be found in [16].…”

Section: 1mentioning

confidence: 99%

“…Wieand has already studied [13,14] the fluctuation of the number of eigenvalues in an arc of the unit circle. Hambly, Keevash, O'Connell and Stark [9] have obtained a central limit theorem for the asymptotic value (in n) of the characteristic polynomial of a permutation matrix, while Zeindler [16] obtained explicit generating functions for those values, even when evaluated inside the unit circle. The present paper simplifies some of the proofs given in the latter paper, but more importantly extends the results in many different ways:…”

mentioning

confidence: 99%

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On averages of randomized class functions on the symmetric groups and their asymptotics

Dehaye

Zeindler

2013

Ann. inst. Fourier

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The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices (n points). In this paper, we generalize many aspects of this situation. We introduce random shifts of the eigenvalues of the permutation matrices, in two different ways: independently or not for each subset of eigenvalues associated to the same cycle. We also consider vastly more general functions than the characteristic polynomial of a permutation matrix, by first finding an equivalent definition in terms of cycle-type of the permutation. We consider other groups than the symmetric group, for instance the alternating group and other Weyl groups. Finally, we compute some asymptotics results when n tends to infinity. This last result requires additional ideas: it exploits properties of the Feller coupling, which gives asymptotics for the lengths of cycles in permutations of many points.

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Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

On averages of randomized class functions on the symmetric groups and their asymptotics

Dehaye

Zeindler

2013

Ann. inst. Fourier

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“…Hambly, Keevash, O-Connell and Stark [9] give a similar result for permutation matrices following the uniform measure. 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: 79%

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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“…We thus reformulate Theorem 2.12. In order to do this, we follow the idea in [11] and [26] and replace [0, 1] d by a slightly smaller set Q such that ϕ ⊂ Q and h|Q is of bounded variation in the sense of Hardy and Krause. We begin with the choice of Q.…”

Section: Function We Call H Of Bounded Variation In the Sense Of Harmentioning

confidence: 99%

“…Then, the characteristic polynomial of M (σ, z) has the same zeros as Z n,z (x). We will study the characteristic polynomial by identifying it with Z n,z (x), following the convention of [7], [26] or [27]. By using that the random variables z i , 1 ≤ i ≤ n are i.i.d., a simple computation shows the following equality in law (see [7], Lemma 4.2):…”

Section: Introductionmentioning

confidence: 99%

The characteristic polynomial of a random permutation matrix at different points

Dang

Zeindler

2014

Stochastic Processes and their Applications

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Abstract. We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show that the behavior at different points is independent in the limit and are asymptotically normal. Our methods enables us to study also the wreath product of permutation matrices and diagonal matrices with iid entries and more general class functions on the symmetric group with a multiplicative structure..

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Permutation Matrices and the Moments of their Characteristics Polynomials

Cited by 19 publications

References 13 publications

On averages of randomized class functions on the symmetric groups and their asymptotics

On averages of randomized class functions on the symmetric groups and their asymptotics

Characteristic polynomials of modified permutation matrices at microscopic scale

The characteristic polynomial of a random permutation matrix at different points

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