logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Next article
Joseph Najnudel; Ashkan Nikeghbali
The distribution of eigenvalues of randomized permutation matrices
(Sur la distribution des valeurs propres de matrices de permutation randomisées)
Annales de l'institut Fourier, 63 no. 3 (2013), p. 773-838, doi: 10.5802/aif.2777
Article PDF | Reviews MR 3137473 | Zbl 1278.15010
Class. Math.: 15A18, 15A52, 20B30
Keywords: Random matrix, permutation matrix, virtual permutation, convergence of eigenvalues

Résumé - 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.

Bibliography

[1] Greg W. Anderson, Alice Guionnet & Ofer Zeitouni, An introduction to random matrices, Cambridge Studies in Advanced Mathematics 118, Cambridge University Press, Cambridge, 2010  MR 2760897 |  Zbl 1184.15023
[2] Richard Arratia, A. D. Barbour & Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2003  MR 2032426 |  Zbl 1040.60001
[3] Patrick Billingsley, Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999, A Wiley-Interscience Publication  MR 1700749 |  Zbl 0172.21201
[4] P-O. Dehaye & D. Zeindler, “On averages of randomized class functions on the symmetric group and their asymptotics”, http://arxiv.org/pdf/0911.4038, 2009
[5] Persi Diaconis, “Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture”, Bull. Amer. Math. Soc. (N.S.) 40 (2003) no. 2, p. 155-178 Article |  MR 1962294 |  Zbl 1161.15302
[6] Persi Diaconis & Mehrdad Shahshahani, “On the eigenvalues of random matrices”, J. Appl. Probab. 31A (1994), p. 49-62, Studies in applied probability Article |  MR 1274717 |  Zbl 0807.15015
[7] Steven N. Evans, “Eigenvalues of random wreath products”, Electron. J. Probab. 7 (2002), p. 1-15 Article |  MR 1902842 |  Zbl 1013.15006
[8] B. M. Hambly, P. Keevash, N. O’Connell & D. Stark, “The characteristic polynomial of a random permutation matrix”, Stochastic Process. Appl. 90 (2000) no. 2, p. 335-346 Article |  MR 1794543 |  Zbl 1047.60013
[9] Serguei Kerov, Grigori Olshanski & Anatoli Vershik, “Harmonic analysis on the infinite symmetric group. A deformation of the regular representation”, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) no. 8, p. 773-778  MR 1218259 |  Zbl 0796.43005
[10] Madan Lal Mehta, Random matrices, Pure and Applied Mathematics (Amsterdam) 142, Elsevier/Academic Press, Amsterdam, 2004  MR 2129906 |  Zbl 1107.15019
[11] F. Mezzadri & N.-C. (eds.) Snaith, Recent perspectives in random matrix theory and number theory, London Mathematical Society Lecture Note Series 322, Cambridge University Press, 2005  MR 2145172 |  Zbl 1065.11002
[12] G. Olshanski, Asymptotic combinatorics with applications to mathematical physics, Lecture Notes in Mathematics 1815, Springer, 2003  MR 2009838
[13] J. Pitman, Combinatorial stochastic processes, Lecture Notes in Mathematics 1875, Springer-Verlag, Berlin, 2006, Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard  MR 2245368 |  Zbl 1103.60004
[14] N. V. Tsilevich, “Distribution of cycle lengths of infinite permutations”, J. Math. Sci. 87 (1997) no. 6, p. 4072-4081 Article |  MR 1374318 |  Zbl 0909.60011
[15] N. V. Tsilevich, “Stationary measures on the space of virtual permutations for an action of the infinite symmetric group”, PDMI Preprint, 1998
[16] Kelly Wieand, “Eigenvalue distributions of random permutation matrices”, Ann. Probab. 28 (2000) no. 4, p. 1563-1587 Article |  MR 1813834 |  Zbl 1044.15017
[17] Kelly Wieand, “Permutation matrices, wreath products, and the distribution of eigenvalues”, J. Theoret. Probab. 16 (2003) no. 3, p. 599-623 Article |  MR 2009195 |  Zbl 1043.60007
[18] Dirk Zeindler, “Permutation matrices and the moments of their characteristic polynomial”, Electron. J. Probab. 15 (2010) no. 34, p. 1092-1118  MR 2659758 |  Zbl 1225.15038
top