Scaling law for the maximum Lyapunov characteristic exponent of infinite product of random matrices

The authors present a simple derivation for the scaling behaviour of the maximum Lyapunov characteristic exponent lambda of infinite product of symplectic random matrices. The considered random matrices depend on a parameter epsilon and lambda =0 for epsilon =0. They obtain lambda varies as epsilon ^beta with either beta =¹/₂ or beta = ²/₃ depending on the probability distribution of the matrix elements. The results are in agreement with a previous numerical simulation.