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 ×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 ∼ M for the rest of the spectrum. It interpolates between the known deterministic behaviour of the Lyapunov exponents for M N (or N fixed) and universal random matrix statistics for M 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.

Products of Many Large Random Matrices and Gradients in Deep Neural Networks

“…Finally, in the specific case where the random matrices X (j) are complex Ginibre matrices (i.e. the matrix entries are iid complex Gaussian), very recent work [2,19] looks at the limiting spectrum under the joint scaling limit d → ∞, n → ∞ where the ratio d/n is fixed or going to ∞. This work analyzes exact determinental formulas for the joint distribution of singular values available in the case of complex Ginibre matrices.…”

“…, n d . Fix 0 ≤ j < j ′ ≤ d. Suppose the weights of N are W (i) , which are drawn iid from the measure µ as in the original definition (2). Then, writing η (j ′ ) ∈ R n j ′ for the n j ′ dimensional ±1-Bernoulli random vector, whose entries are independent and take the values ±1 with probability 1/2, we have…”

“…, n d and randomly initialized weights drawn i.i.d. from the measure µ and scaled to have variance 2/fan-in = 2/n i as in (2). Suppose also that the biases of N are drawn i.i.d.…”

“…The purpose of this article is to prove several new results about such products in the regime where the number of terms and the sizes of matrices grow simultaneously. This regime has attracted attention [2,19] but remains poorly understood. We find new phenomena not present when the number of terms and the size of the matrices are sent to infinity sequentially rather than simultaneously (see Section 1.1 for more on this point).…”

“…Let M (d) = X (d) · · · X (1) be as in (2). Then, the norm of the vector M (d) u is approximately log-normal distributed:…”