On fluctuations of eigenvalues of random permutation matrices

Ben Arous, Gérard; Dang, Kim

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 620-647.

Les statistiques linéaires d’observables régulières du spectre de matrices de permutations, choisies aléatoirement sous une distribution générale de Ewens, donnent lieu à un phénomène intéressant de non-universalité. Bien qu’elles aient une variance bornée, leurs fluctuations ne sont pas asymptotiquement Gaussiennes, mais infiniment divisibles. Si l’observable est moins régulière, la variance diverge et les fluctuations sont Gaussiennes. Le degré de régularité est mesuré en termes de la qualité de l’approximation trapézoidale de l’intégrale de l’observable.

Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting non-universality phenomenon. Though they have bounded variance, their fluctuations are asymptotically non-Gaussian but infinitely divisible. The fluctuations are asymptotically Gaussian for less smooth linear statistics for which the variance diverges. The degree of smoothness is measured in terms of the quality of the trapezoidal approximations of the integral of the observable.

Classification : 60F05, 15B52, 60B20, 60B15, 60C05, 60E07, 65D30
Mots clés : random matrices, linear eigenvalue statistics, random permutations, infinitely divisible distributions, trapezoidal approximations

[1] R. Arratia, A. Barbour and S. Tavaré. Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 (3) (1992) 519–535. | MR | Zbl

[2] R. Arratia, A. Barbour and S. Tavaré. Logarithmic Combinatorial Structures: A Probabilistic Approach. EMS Monographs in Mathematics. EMS, Zürich, 2003. | DOI | MR | Zbl

[3] R. Arratia and S. Tavaré. Limit theorems for combinatorial structures via discrete process approximations. Random Structures Algorithms 3 (3) (1992) 321–345. | MR | Zbl

[4] R. Arratia and S. Tavaré. The cycle structure of random permutations. Ann. Probab. 20 (3) (1992) 1567–1591. | MR | Zbl

[5] Z. D. Bai. Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 (3) (1999) 611–677. | MR | Zbl

[6] Z. D. Bai and J. W. Silverstein. CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 (2004) 533–605. | MR | Zbl

[7] A. Barbour and S. Tavaré. A rate for the Erdös–Turan law. Combin. Probab. Comput. 3 (1994) 167–176. | DOI | MR | Zbl

[8] A. Boutet De Monvel, L. Pastur and M. Shcherbina. On the statistical mechanics approach in the random matrix theory: Integrated density of states. J. Stat. Phys. 79 (1995) 585–611. | DOI | MR | Zbl

[9] B. Büttgenbach, G. Lüttgens and R. Nessel. On some problems concerning best constants for the midpoint and trapezoidal rule. In General Inequalities 6 (Oberwolfach, 1990) 393–409. Internat. Ser. Numer. Math. 103. Birkhäuser, Basel, 1992. | MR | Zbl

[10] S. Chatterjee. Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 (1–2), (2009) 1–40. | MR | Zbl

[11] O. Costin and J. Lebowitz. Gaussian fluctuation in random matrices. Phys. Rev. Lett. 75 (1995) 69–72. | DOI | MR

[12] D. Cruz-Uribe and C. J. Neugebauer. Sharp error bounds for the trapezoidal rule and Simpson’s rule. J. Inequal. Pure Appl. Math. 3 (4), (2002) Article 49, 1–22. | EuDML | MR | Zbl

[13] P. J. Davis and P. Rabinowitz. Methods of Numerical Integration, 2nd edition. Academic Press, New York, 1984. | MR | Zbl

[14] P. Diaconis. Patterns in eigenvalues: The 70th Josiah Willard Gibbs lecture. Bull. Amer. Math. Soc. (N.S.) 40 (2) (2003) 155–178. | MR | Zbl

[15] P. Diaconis and S. N. Evans. Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 (2001) 2615–2633. | DOI | MR | Zbl

[16] P. Diaconis and M. Shahshahani. On the eigenvalues of random matrices. J. Appl. Probab. 31 (A) (1994) 49–61. | MR | Zbl

[17] I. Dumitriu and A. Edelman. Global spectrum fluctuations for the Hermite and Laguerre ensembles via matrix models. J. Math. Phys. 47 (6), 063302 (2006) 1–36. | MR | Zbl

[18] I. Dumitriu and S. Pal. Sparse regular random graphs: Spectral density and eigenvectors. Ann. Probab. 40 (5) (2012) 2197–2235. | MR | Zbl

[19] I. Dumitriu, T. Johnson, S. Pal and E. Paquette. Functional limit theorems for random regular graphs. Probab. Theory Related Fields 156 (2013) 921–975. | DOI | MR | Zbl

[20] W. J. Ewens. The sampling theory of selectively neutral alleles. Theoret. Population Biology 3 (1972) 87–112. | DOI | MR | Zbl

[21] B. M. Hambly, P. Keevash, N. O’Connell and D. Stark. The characteristic polynomial of a random permutation matrix. Stochastic Process. Appl. 90 (2000) 335–346. | DOI | MR | Zbl

[22] E. Hille and O. Szasz. On the completeness of Lambert functions. Bull. Amer. Math. Soc. (N.S.) 42 (1936) 411–418. | JFM | MR

[23] C. Hughes, J. Najnudel, A. Nikeghbali and D. Zeindler. Random permutation matrices under the generalized Ewens measure. Ann. Appl. Probab. 20 (3) (2013) 987–1024. | MR | Zbl

[24] K. Johansson. On random matrices from the compact classical groups. Ann. of Math. (2) 145 (1997) 519–545. | MR | Zbl

[25] K. Johansson. On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 (1) (1998) 151–204. | MR | Zbl

[26] D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 (1982) 1–38. | DOI | MR | Zbl

[27] A. M. Khorunzhy, B. A. Khoruzhenko and L. A. Pastur. Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37 (1996) 5033–5060. | DOI | MR | Zbl

[28] J. H. Loxton and J. W. Sanders. The kernel of a rule of approximate integration. J. Austral. Math. Soc. Ser. B 21 (1980) 257–267. | DOI | MR | Zbl

[29] A. Lytova and L. Pastur. Central limit theorem for linear eigenvalue statistics of random matrices with independent entries. Ann. Probab. 37 (2009) 1778–1840. | DOI | MR | Zbl

[30] L. Pastur. Limiting laws of linear eigenvalue statistics for unitary invariant matrix models. J. Math. Phys. 47 (2006) 103303. | MR | Zbl

[31] Q. Rahman and G. Schmeisser. Characterization of the speed of convergence of the trapezoidal rule. Numer. Math. 57 (1990) 123–138. | DOI | EuDML | MR | Zbl

[32] B. Rider and J. W. Silverstein. Gaussian fluctuations for non-Hermitian random matrix ensembles. Ann. Probab. 34 (6) (2006) 2118–2143. | MR | Zbl

[33] L. A. Shepp and S. P. Lloyd. Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 (2) (1966) 340–357. | MR | Zbl

[34] S. R. S. Varadhan. Probability Theory. Courant Lecture Notes 7. Amer. Math. Soc., Providence, RI, 2001. | DOI | MR | Zbl

[35] G. A. Watterson. Models for the logarithmic species abundance distributions. Theoret. Population Biology 6 (1974) 217–250. | DOI | MR | Zbl

[36] K. Wieand. Eigenvalue distributions of random permutation matrices. Ann. Probab. 28 (4) (2000) 1563–1587. | MR | Zbl

[37] K. Wieand. Eigenvalue distributions of random unitary matrices. Probab. Theory Related Fields 123 (2002) 202–224. | DOI | MR | Zbl

[38] A. Zygmund. Trigonometric Series, Vols. I, II, 3rd edition. Cambridge Univ. Press, Cambridge, 2002. | MR | Zbl