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Power law scaling of the top Lyapunov exponent of a Product of Random Matrices

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Abstract

A sequence of i.i.d. matrix-valued random variables\(\left\{ {X_n } \right\} \cdot X_n = \left( {\begin{array}{*{20}c} 1 & d \\ 0 & 1 \\ \end{array} } \right)\) with probabilityp and\(X_n = \left( {\begin{array}{*{20}c} {1 + a(\varepsilon )} & {b(\varepsilon )} \\ {c(\varepsilon )} & {1 + a(\varepsilon )} \\ \end{array} } \right)\) with probability 1−p is considered. Leta(ε) = a 0 ε + O(ε), c(ε) = c 0 ε + O(ε) lim ε→0 b(ε) = Oa 0,c 0, ε>0, andb(ε)>0 for all ε>0. It is shown show that the top Lyapunov exponent of the matrix productX n X n-1...X 1, λ = limn → ∞ (1/n) ∣nX n X n-1...X 1∥ satisfies a power law with an exponent 1/2. That is, limε → 0(ln λ/ln ε) = 1/2.

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Ravishankar, K. Power law scaling of the top Lyapunov exponent of a Product of Random Matrices. J Stat Phys 54, 531–537 (1989). https://doi.org/10.1007/BF01023493

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  • DOI: https://doi.org/10.1007/BF01023493

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