Products of M i.i.d. random matrices of size N relate classical limit theorems in Probability Theory (large M and N=1) to Lyapunov exponents in Dynamical Systems (large M and finite N), and to universality in Random Matrix Theory (finite M and large N). Under the two different limits of large M and large N, the  eigenvalue statistics for the random matrix product display Gaussian and  RMT  universality, respectively. However, what happens if both M and N go to infinity simultaneously? This problem lies at the heart of understanding two kinds of universal limits.  In this talk we examine it and investigate possible phase transition and critical phenomena.

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