Advertisement

Communications in Mathematical Physics

, Volume 143, Issue 3, pp 591–598 | Cite as

The triangle law for Lyapunov exponents of large random matrices

  • Marco Isopi
  • Charles M. Newman
Article

Abstract

For products,A(t)·A(t−1)...A(1), of i.i.d.N×N random matrices, with i.i.d. entries, a triangle law governs theN→∞ distribution of Lyapunov exponents, much like Wigner's quarter-circle law governs the singular values ofA(1). Our proof requires finite fourth moments and a bounded density; the result was previously derived only in the Gaussian case.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Lyapunov Exponent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B] Ball, K.: Cube slicing inR n. Proc. Am. Math. Soc.97, 465–473 (1986)Google Scholar
  2. [BL] Bourgerol, P., Lacroix, J.: Products of random matrices with applications to Schrödinger operators. Boston: Birkhäuser 1985Google Scholar
  3. [BYK] Bai, Z.D., Yin, Y.Q., Krishnaiah, P.R.: On the limiting empirical distribution function of the eigenvalues of a multivariateF matrix. Theory Prob. Appl.32, 490–500 (1987)CrossRefGoogle Scholar
  4. [CN] Cohen, J.E., Newman, C.M.: The stability of large random matrices and their products. Ann. Prob.12, 283–310 (1984)Google Scholar
  5. [F] Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc.108, 377–428 (1963)Google Scholar
  6. [FK] Furstenberg, H., Kifer, Y.: Random matrix products and measures on projective spaces. Ist. J. Math.46, 12–32 (1983)Google Scholar
  7. [GR] Guivarc'h, Y., Raugi, A.: Frontière de Fürstenberg, propriétés de contraction et theorèmes de convergence. Zeit. Wahrscheinlichkeitstheorie Verw. Gebiete69, 187–242 (1985)Google Scholar
  8. [H] Hensley, D.: Slicing the cube inR n and probability (bounds for the measure of a central cube slice inR n by probability methods). Proc. Am. Math. Soc.73, 95–100 (1979)Google Scholar
  9. [MP] Marčenko, V.A., Pastur, L.A.: Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sb.1, 457–483 (1967)Google Scholar
  10. [N1] Newman, C.M.: The distribution of Lyapunov exponents: exact results for random matrices. Commun. Math. Phys.103, 121–126 (1986)CrossRefGoogle Scholar
  11. [N2] Newman, C.M.: Lyapunov exponents for some products of random matrices: exact expressions and asymptotic distributions. In: Random matrices and their applications, Cohen, J.E., Kesten, H., Newman, C.M. (eds.). Providence, RI: AMS 1986Google Scholar
  12. [O] Oseledec, V.I.: A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc.19, 197–231 (1968)Google Scholar
  13. [R] Raghunathan, M.S.: A proof of Oseledec's multiplicative ergodic theorm. Isr. J. Math.32, 356–362 (1979)Google Scholar
  14. [Ro] Rogozin, B.A.: Estimation of the maximum of a convolution of bounded densities. Theory Prob. Appl.32, 48–56 (1987)CrossRefGoogle Scholar
  15. [W] Wigner, E.P.: Random matrices in physics. SIAM Rev.9, 1–23 (1967)CrossRefGoogle Scholar
  16. [Wi] Wilks, S.S.: Mathematical statistics. New York: Wiley 1962Google Scholar
  17. [Y] Yin, Y.Q.: Limiting spectral distribution for a class of random matrices. J. Multivar. Anal.20, 50–68 (1986)CrossRefGoogle Scholar
  18. [YBK] Yin, Y.Q., Bai, Z.D., Krishnaiah, P.R.: On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Relat. Fields78, 509–521 (1988)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Marco Isopi
    • 1
  • Charles M. Newman
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew YorkUSA

Personalised recommendations