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


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.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Lyapunov Exponent 
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  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

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