Journal of Statistical Physics

, Volume 166, Issue 1, pp 24–38 | Cite as

Infinite Products of Random Isotropically Distributed Matrices

  • A. S. Il’yn
  • V. A. Sirota
  • K. P. Zybin


Statistical properties of infinite products of random isotropically distributed matrices are investigated. Both for continuous processes with finite correlation time and discrete sequences of independent matrices, a formalism that allows to calculate easily the Lyapunov spectrum and generalized Lyapunov exponents is developed. This problem is of interest to probability theory, statistical characteristics of matrix T-exponentials are also needed for turbulent transport problems, dynamical chaos and other parts of statistical physics.


Lyapunov exponents Random matrices T-exponential Functional integral Stochastic equations Turbulence Qft 



This work is supported by the RAS program ‘Nonlinear dynamics in mathematical and physical sciences’.


  1. 1.
    Crisanti, A., Paladin, G., Vulpiani, A.: Products of random matrices in statistical physics. Springer, New York (1993)CrossRefMATHGoogle Scholar
  2. 2.
    Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Oseledec, V.I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 179–210 (1968). English translation, Trans. Moscow Math. Soc. 19, 197-221 (1968)MathSciNetGoogle Scholar
  4. 4.
    Tutubalin, V.N.: A variant of the local limit theorem for products of random matrices. Theor. Probab. Appl. 22(2), 203–214 (1978)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Letchikov, A.V.: Products of unimodular independent random matrices. Russ. Math. Surv. 51, 49–96 (1996)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Gantmacher, F.R.: Matrix theory, 2nd edn. AMS Providence, Rhode Island (1990)Google Scholar
  7. 7.
    Le Jan, Y.: On isotropic Brownian motions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 609–620 (1985)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Baxendale, P.H.: The Lyapunov spectrum of a stochastic flow of diffeomorphisms in Lyapunov exponents, Bremen 1984. In: Arnold, L., Wihstutz, V. (eds.) Lecture Notes in Math, vol. 1186, pp. 322–337. Springer, New York (1986)Google Scholar
  9. 9.
    Gamba, A., Kolokolov, I.: The Lyapunov spectrum of a continuous product of random matrices. J. Stat. Phys. 85, 489–499 (1996)ADSCrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Gamba, A.: Finite-time Lyapunov exponents for products of random transformations. J. Stat. Phys. 112(1–2), 193–218 (2003)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Zybin, K.P., Sirota, V.A.: Multifractal structure of fully developed turbulence. Phys. Rev. E 88(4), 043017 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Zybin, K.P., Sirota, V.A.: Model of stretching vortex filaments and foundations of the statistical theory of turbulence. Sov. Phys. Uspekhi 58(6), 556–573 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    Balkovsky, E., Fouxon, A.: Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem. Phys. Rev. E 60, 4164–4174 (1999)ADSCrossRefMathSciNetGoogle Scholar
  14. 14.
    Falkovich, G., Gawedzki, K., Vergassola, M.: Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975 (2001)ADSCrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Zeldovich, Y.B., Ruzmaikin, A.A., Molchanov, S.A., Sokolov, D.D.: Kinematic dynamo problem in a linear velocity field. J. Fluid Mech. 144, 1–11 (1984)ADSCrossRefGoogle Scholar
  16. 16.
    Chertkov, M., Falkolovich, G., Kolokolov, I., Vergassola, M.: Small-scale turbulent dynamo. Phys. Rev. Lett. 83, 4065–4068 (1999)ADSCrossRefGoogle Scholar
  17. 17.
    Ooi, A., Martin, J., Soria, J., Chong, M.S.: A study of the evolution and characteristics of the invariants of the velocity gradient tensor in isotropic turbulence. J. Fluid Mech. 381, 141–174 (1999)ADSCrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Kolmogorov, A.N.: Dissipation of energy in the locally isotropic turbulence. Proc. Math. Phys. Sci. 434, 15–17 (1991)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Frisch, U.: Turbulence: the legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  20. 20.
    Girimaji, S.S., Pope, S.B.: Material-element deformation in isotropic turbulence. J. Fluid Mech. 220, 427–458 (1990)ADSCrossRefGoogle Scholar
  21. 21.
    Il’yn, A.S., Sirota, V.A., Zybin, K.P.: Statistical properties of the T-Exponential of isotropically distributed random matrices. J. Stat. Phys. 163(4), 765–783 (2016)ADSCrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Klyatskin, V.I.: Dynamics of stochastic systems. Elsevier, Amsterdam (2005)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.P.N.Lebedev Physical Institute of RASMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia

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