# The distribution of eigenvalues of randomized permutation matrices

Joseph Najnudel^{[1]}; Ashkan Nikeghbali^{[1]}

- [1] Universität Zürich Institut für Mathematik Winterthurerstrasse 190 8057-Zürich( Switzerland)

Annales de l’institut Fourier (2013)

- Volume: 63, Issue: 3, page 773-838
- ISSN: 0373-0956

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topNajnudel, Joseph, and Nikeghbali, Ashkan. "The distribution of eigenvalues of randomized permutation matrices." Annales de l’institut Fourier 63.3 (2013): 773-838. <http://eudml.org/doc/275490>.

@article{Najnudel2013,

abstract = {In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter $\theta >0$) by replacing the entries equal to one by more general non-vanishing complex random variables. For these ensembles, in contrast with more classical models as the Gaussian Unitary Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very explicitly computed by using the cycle structure of the permutations. Moreover, by using the so-called virtual permutations, first introduced by Kerov, Olshanski and Vershik, and studied with a probabilistic point of view by Tsilevich, we are able to define, on the same probability space, a model for each dimension greater than or equal to one, which gives a meaning to the notion of almost sure convergence when the dimension tends to infinity. In the present paper, depending on the precise model which is considered, we obtain a number of different results of convergence for the point measure of the eigenvalues, some of these results giving a strong convergence, which is not common in random matrix theory.},

affiliation = {Universität Zürich Institut für Mathematik Winterthurerstrasse 190 8057-Zürich( Switzerland); Universität Zürich Institut für Mathematik Winterthurerstrasse 190 8057-Zürich( Switzerland)},

author = {Najnudel, Joseph, Nikeghbali, Ashkan},

journal = {Annales de l’institut Fourier},

keywords = {Random matrix; permutation matrix; virtual permutation; convergence of eigenvalues; random matrix},

language = {eng},

number = {3},

pages = {773-838},

publisher = {Association des Annales de l’institut Fourier},

title = {The distribution of eigenvalues of randomized permutation matrices},

url = {http://eudml.org/doc/275490},

volume = {63},

year = {2013},

}

TY - JOUR

AU - Najnudel, Joseph

AU - Nikeghbali, Ashkan

TI - The distribution of eigenvalues of randomized permutation matrices

JO - Annales de l’institut Fourier

PY - 2013

PB - Association des Annales de l’institut Fourier

VL - 63

IS - 3

SP - 773

EP - 838

AB - In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter $\theta >0$) by replacing the entries equal to one by more general non-vanishing complex random variables. For these ensembles, in contrast with more classical models as the Gaussian Unitary Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very explicitly computed by using the cycle structure of the permutations. Moreover, by using the so-called virtual permutations, first introduced by Kerov, Olshanski and Vershik, and studied with a probabilistic point of view by Tsilevich, we are able to define, on the same probability space, a model for each dimension greater than or equal to one, which gives a meaning to the notion of almost sure convergence when the dimension tends to infinity. In the present paper, depending on the precise model which is considered, we obtain a number of different results of convergence for the point measure of the eigenvalues, some of these results giving a strong convergence, which is not common in random matrix theory.

LA - eng

KW - Random matrix; permutation matrix; virtual permutation; convergence of eigenvalues; random matrix

UR - http://eudml.org/doc/275490

ER -

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