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

Abstract

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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 θ > 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.

How to cite

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Najnudel, 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 $\theta &gt;0$) 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 $\theta &gt;0$) 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 -

References

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