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

Abstract

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.

7 pages, 4 figures; v2: map to Dyson's Brownian motion added and typo corrected in the soft-edge kernel; v3: added some description on derivations

References in corpus (1)

• The structure and dynamics of multilayer networks

Cited by in corpus (9)

• Products of Many Large Random Matrices and Gradients in Deep Neural Networks
• Limits and fluctuations of $p$-adic random matrix products
• Random-matrix models of monitored quantum circuits
• Universality of local spectral statistics of products of random matrices
• The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$
• Averages of products and ratios of characteristic polynomials in polynomial ensembles
• Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions
• Hard Edge Statistics of Products of Pólya Ensembles and Shifted GUE's
• Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance