From Integrable to Chaotic Systems: Universal Local Statistics of Lyapunov exponents

Systems where time evolution follows a multiplicative process are ubiquitous in physics. We study a toy model for such systems where each time step is given by multiplication with an independent random $N\times N$ matrix with complex Gaussian elements, the complex Ginibre ensemble. This model allows to explicitly compute the Lyapunov exponents and local correlations amongst them, when the number of factors $M$ becomes large. While the smallest eigenvalues always remain deterministic, which is also the case for many chaotic quantum systems, we identify a critical double scaling limit $N\sim M$ for the rest of the spectrum. It interpolates between the known deterministic behaviour of the Lyapunov exponents for $M\gg N$ (or $N$ fixed) and universal random matrix statistics for $M\ll N$ (or $M$ fixed), characterising chaotic behaviour. After unfolding this agrees with Dyson's Brownian Motion starting from equidistant positions in the bulk and at the soft edge of the spectrum. This universality statement is further corroborated by numerical experiments, multiplying different kinds of random matrices. It leads us to conjecture a much wider applicability in complex systems, that display a transition from deterministic to chaotic behaviour.