The product of $m$ real $N \times N$ Ginibre matrices:\ABScr Real eigenvalues in the critical regime $m=O(N)$

The study of products of random matrices has been proposed many decades ago by Bellman and by Furstenberg and Kesten with the motivation to understand the properties of the Lyapunov exponents in this toy model for chaotic dynamical systems.
In this talk, I will discuss the real eigenvalues of products of random matrices with i.i.d. Gaussian entries. In the critical regime where the size of matrices and the number of products are proportional in the large system, I will present the mean and variance of the number of real eigenvalues. Furthermore, in the Lyapunov scaling, I will introduce the densities of real eigenvalues, which interpolates Ginibre's circular law with Newman's triangular law.
This is based on joint work with Gernot Akemann.