Lyapunov Exponent, Universality and Phase Transition for Products of Random Matrices

Abstract

Products of M i.i.d. random matrices of size \(N \times N\) are related to classical limit theorems in probability theory (\(N=1\) and large M), to Lyapunov exponents in dynamical systems (finite N and large M), and to universality in random matrix theory (finite M and large N). Under the two different limits of \(M\rightarrow \infty \) and \(N\rightarrow \infty \), the local singular value statistics display Gaussian and random matrix theory universality, respectively. However, it is unclear what happens if both M and N go to infinity. This problem, proposed by Akemann et al. (J Phys A 47(39):395202, 2014) and Deift (SIGMA Symmetry Integr Geom Methods Appl 13, 2017), lies at the heart of understanding both kinds of universal limits. In the case of complex Gaussian random matrices, we prove that there exists a crossover phenomenon as the relative ratio of M and N changes from 0 to \(\infty \): sine and Airy kernels from the Gaussian Unitary Ensemble (GUE) when \(M/N \rightarrow 0\), Gaussian fluctuation when \(M/N \rightarrow \infty \), and new critical phenomena when \(M/N \rightarrow \gamma \in (0,\infty )\). Accordingly, we further prove that the largest singular value undergoes a phase transition between the Gaussian and GUE Tracy–Widom distributions.