Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 9 (2004) > Paper 9 open journal systems 


Poisson Statistics for the Largest Eigenvalues of Wigner Random Matrices with Heavy Tails

Alexander Soshnikov, University of California at Davis, USA


Abstract
We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson


Full text: PDF

Pages: 82-91

Published on: August 24, 2004


Bibliography
  1. L.Arnold. On the asymptotic distribution of the eigenvalues of random matrices. J. Math. Anal. Appl. 20 (1967), 262-268. Math. Review MR0217833
  2. Z.Burda et. al., Free random L'{e}vy matrices, Phys Rev E 65 (2002), 021106. Math. Review MR1908289
  3. M.V.Berry, M. Tabor. Level clustering in the regular spectrum. Proc. R. Soc. London Ser. A 356, (1977), 375-394. Math. Review number not available.
  4. P.Cizeau, J.P.Bouchaud. Theory of L'{e}vy matrices. Phys Rev E 50, (1994),1810-1822. Math. Review number not available.
  5. Z.Cheng, J.L.Lebowitz and P.Major. On the number of lattice points between two enlarged and randomly shifted copies of an oval. Prob. Theo. Rel. Fields 100, (1994), 253-268. Math. Review 95j:60037
  6. D.J. Daley, D.Vere-Jones. An Introduction to the Theory of Point Processes, Vol.I, Elementary theory and methods. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2003. xxii+469 pp. ISBN 0-387-95541-0 Math. Review 2004c:60001
  7. P.Deift. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, Vol. 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. viii+273 pp. ISBN: 0-9658703-2-4; 0-8218-2695-6. Math. Review 2000g:47048
  8. Z. F"{u}redi and J. Koml'os. The eigenvalues of random symmetric matrices. Combinatorica 1 (1981), 233-241. Math. Review 83e:15010
  9. P. Forrester. The spectral edge of random matrix ensembles. Nucl. Phys. B 402(1994), 709-728. Math. Review 94h:82031
  10. V.L.Girko{cyr Spektralcprime naya teoriya sluchaui nykh matrits}. (Russian) [Spectral theory of random matrices] {cyr Teoriya Veroyatnosteui i Matematicheskaya Statistika}. [Probability Theory and Mathematical Statistics] ``Nauka'', Moscow, 1988. 376 pp. ISBN: 5-02-013749-9 MR0955497 (90g:60034) Math. Review 90g:60034
  11. I.A.Ibragimov, Yu.V.Linnik. Independent and Stationary Sequences of Random Variables, translation from the Russian edited by J.F.C.Kingman, Wolters-Noordhoff Publishing, Groningen, 1971. Math. Review MR0322926
  12. R.A.Janik. New multicritical random matrix ensembles. Nuclear Phys B635, (2002),492-504. Math. Review 2003i:82045
  13. K.Johansson. Universality of the Local Spacing Distribution in Certain Ensembles of Hermitian Wigner Matrices. Commun. Math. Phys.215, (2001), 683-705. Math. Review 2002j:15024
  14. J. Karamata. Sur un mode de croissance r'{e}guli`{e}re des fonctions. Mathematica (Cluj)4 (1930), 38-53. Math. Review number not available.
  15. M.R. Leadbetter, G.Lindgren and H. Rootz'{e}n. Extremes and related properties of random sequences and processes. Springer Series in Statistics. Springer-Verlag, New York-Berlin, 1983. xii+336 pp. ISBN: 0-387-90731-9 Math. Review 84h:60050
  16. P.Major. Poisson law for the number of lattice points in a random strip with finite area. Prob. Theo. Rel. Fields 92 (1992), 423-464. Math. Review 93g:60021
  17. V.A. Marchenko, L.A. Pastur. Distribution of some sets of random matrices. Math. USSR-Sb.1 (1967),457-483. Math. Review number not available.
  18. J. Marklof. Pair correlation densities of inhomogeneous quadratic forms. Annals of Mathematics 158, (2003), 419-471. Math. Review MR2018926
  19. J. Marklof. Pair correlation densities of inhomogeneous quadratic forms II. Duke Mathematical Journal115, (2002), 409-434. Math. Review 2004f:11110a
  20. J. Marklof. The Berry-Tabor conjecture. Proceedings of the 3rd European Congress of Mathematics, Barcelona 2000, Progress in Mathematics 202, (2001), 421-427. Math. Review MR1905381
  21. M.L.Mehta. Random Matrices. Academic Press, New York, 1991. Math. Review 92f:82002
  22. N.Minami. Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys.177, (1996), 709-725. Math. Review 97d:82046
  23. S.A.Molchanov. The local structure of the spectrum of the one-dimensional Schr"{o}dinger operator. Commun. Math. Phys.78 (1981), 429-446. Math. Review 82d:35076
  24. L.A. Pastur. On the spectrum of random matrices. Teor. Mat. Fiz.10, (1972),102-112. Math. Review MR0475502
  25. P.Sarnak. Values at integers of binary quadratic forms. Harmonic Analysis and Number Theory, Montreal, PQ, 1996, 181-203, CMS Conf. Proc. 21, Amer. Math. Soc., Providence, RI, 1997. Math. Review 98j:11024
  26. E.Seneta. Regularly Varying Functions. Lecture Notes in Mathematics 508 (eds. A.Dold and B.Eckmann), Springer, New York, 1976. Math. Review MR0453936
  27. Ya.Sinai. Poisson distribution in a geometric problem. Adv. Sov. Math., AMS Publ.3, (1991), 199-215. Math. Review 92i:60024
  28. A.Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys.207, (1999), 697-733. Math. Review 2001i:82037
  29. A.Soshnikov. A Note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J. Stat. Phys.108, (2002), 1033-1056. Math. Review 2003h:62108
  30. A.Soshnikov, Y.Fyodorov. On the singular values of random matrices with independent Cauchy entries. arXiv preprint math.PR/0403425. Math. Review number not available.
  31. C.A.Tracy, H.Widom. Level-spacing distribution and the Airy kernel. Commun. Math. Phys. 159, (1994), 151-174. Math. Review 95e:82003
  32. C.A.Tracy, H.Widom. On orthogonal and symplectic random matrix ensembles. Commun. Math. Phys.177 (1996), 724-754. Math. Review 97a:82055
  33. E.Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math.62, (1955),548-564. Math. Review 17,1097c
  34. E.Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math.67, (1958),325-328. Math. Review MR0095527
  35. E.Wigner. Random matrix theory in physics. SIAM Rev.9, (1967), 1-23. Math. Review number not available.
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X