Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 13 (2008) > Paper 28 open journal systems 


On the lower bound of the spectral norm of symmetric random matrices with independent entries

Sandrine Peche, Institut Fourier, Grenoble, France
Alexander Soshnikov, University of California at Davis, USA


Abstract
We show that the spectral radius of an N-dimensional random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by 2σ - o(N-6/11+ε), where σ2 is the the variance of the matrix entries and ε > 0 is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any ε > 0 one has ||AN|| = 2σ + o(N-6/11+ε) with probability going to 1 as N goes to infinity.


Full text: PDF

Pages: 280-290

Published on: June 1, 2008
Bibliography
  1. N.Alon, M. Krivelevich, and V.Vu, On the concentration of eigenvalues of random symmetric matrices. Israel J. Math. 131 (2002), 259-267. Math. Review MR1942311
  2. L.Arnold. On the asymptotic distribution of the eigenvalues of random matrices. J. Math. Anal. Appl. 20 (1967), 262-268. Math. Review MR0217833
  3. 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
  4. Z. F"{u}redi and J. Koml'os. The eigenvalues of random symmetric matrices. Combinatorica 1 (1981), 233-241. Math. Review 83e:15010
  5. A. Guionnet and O. Zeitouni. Concentration of the spectral measure for large matrices. Electron. Comm. Probab. 5 (2000), 119-136. Math. Review 2001k:15035
  6. M. Ledoux. The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001. x+181 pp. ISBN 0-8218-2864-9 Math. Review 2003k:28019
  7. M. Ledoux. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, 120--216, Lecture Notes in Math., 1709, Springer, Berlin, 1999. Math. Review 2002j:60002
  8. S.Péché and A. Soshnikov. Wigner random matrices with non-symmetrically distributed entries. J. Stat. Phys. 129 (2007), 857-884.
  9. Y. Sinai and A. Soshnikov. Central limit theorem for traces of large random symmetric matrices with independent matrix elements. Bol. Soc. Brasil. Mat. (N.S.)29 (1998), 1-24. Math. Review 99f:60053
  10. Y. Sinai and A. Soshnikov. A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. Funct. Anal. Appl.32 (1998), 114-131. Math. Review 2000c:82041
  11. A.Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys.207, (1999), 697-733. Math. Review 2001i:82037
  12. M. Talagrand. Concentration of measures and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publications Mathematiques 81, (1996), 73-205. Math. Review 97h:60016
  13. C.A.Tracy, H.Widom. Level-spacing distribution and the Airy kernel. Commun. Math. Phys. 159, (1994), 151-174. Math. Review 95e:82003
  14. C.A.Tracy, H.Widom. On orthogonal and symplectic random matrix ensembles. Commun. Math. Phys.177 (1996), 724-754. Math. Review 97a:82055
  15. V.H. Vu. Spectral norm of random matrices. STOC'05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing. 423--430, ACM, New York, 2005. Math. Review 2006h:15025
  16. E.Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math.62, (1955),548-564. Math. Review 17,1097c
  17. E.Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math.67, (1958),325-328. Math. Review MR0095527
















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