Lyapunov exponent, universality and phase transition for products of random matrices

Dang-Zheng Liu, Dong Wang, Yanhui Wang

Published 2018-09-30Version 1

We are devoted to solving one problem, which is proposed by Akemann, Burda, Kieburg \cite{ABK} and Deift \cite{D17}, on local statistics of finite Lyapunov exponents for $M$ products of $N\times N$ Gaussian random matrices as both $M$ and $N$ go to infinity. When the ratio $(M+1)/N$ changes from $0$ to $\infty$, we prove that the local statistics undergoes a transition from GUE to Gaussian. Especially at the critical scaling $(M+1)/N \to \gamma \in (0,\infty)$, we observe a phase transition phenomenon.