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

AbstractA 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) = a0ε + O(ε), c(ε) = c0ε + O(ε) limε→0b(ε) = Oa0,c0, ε>0, andb(ε)>0 for all ε>0. It is shown show that the top Lyapunov exponent of the matrix productXnXn-1...X1, λ = limn → ∞ (1/n) ∣n ∥XnXn-1...X1∥ satisfies a power law with an exponent 1/2. That is, limε → 0(ln λ/ln ε) = 1/2.