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The largest Lyapunov exponent for random matrices and directed polymers in a random environment. (English) Zbl 0673.60066
The paper establishes bounds for the largest Lyapunov exponent of random matrix products with matrices being either d-dimensional Laplacians with random entries or symplectic matrices which arise in the study of d- dimensional lattices of coupled, nonlinear oscillators. The method uses the expression for the largest Lyapunov exponent via a random walk in a random environment.
Reviewer: Y.Kifer

60H25 Random operators and equations (aspects of stochastic analysis)
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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[1] Bouchard, J.P., Georges, A., Hansel, D., Le Doussal, P., Maillard, J.M.: J. Phys. A19, L1452-L1152 (1986)
[2] Cohen, J.E., Newman, C.M.: The stability of large random matrices and their products. Ann. Prob.12, 283-310 (1984) · Zbl 0543.60098
[3] Eckmann, J.-P., Wayne, C.E.: Liapunov spectra for infinite chains of non-linear oscillators. J. Stat. Phys.50, 853-878 (1987) · Zbl 1084.37500
[4] Imbrie, J.Z., Spencer, T.: Directed polymers in a random environment (preprint) · Zbl 1084.82595
[5] Livi, R., Politi, A., Ruffo, S.: Distribution of characteristic exponents in the thermodynamic limit. J. Phys. A19, 2033-2040 (1986) · Zbl 0624.58030
[6] Newman, C.M.: The distribution of Liapunov exponents. Commun. Math. Phys.103, 121-126 (1986) · Zbl 0593.58051
[7] Newman, C.M.: Liapunov exponents for some products of random matrices: exact expressions and asymptotic distributions. In: Random matrices and their applications,? Vol. 50, p. 121. Cohen, J.E., Kesten, H., Newman, C.M. (eds.). Providence, RI: AMS Contemporary Mathematics 1986 · Zbl 0584.60018
[8] Paladin, G., Vulpiani, A.: Scaling law and asymptotic distribution of Liapunov exponents in conservative dynamical systems with many degrees of freedom. J. Phys. A19, 1881-1888 (1986) · Zbl 0609.70026
[9] Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. IHES50, 275-306 (1979) · Zbl 0426.58014
[10] Spitzer, F.: Principles of random walk, 2nd edn. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0359.60003
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