Products of M i.i.d. random matrices of size NΓN 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 Mββ and Nββ, 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, Burda, Kieburg \cite{Akemann-Burda-Kieburg14} and
Deift \cite{Deift17}, 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 β: sine and Airy kernels from the Gaussian Unitary
Ensemble (GUE) when M/Nβ0, Gaussian fluctuation when M/Nββ,
and new critical phenomena when M/NβΞ³β(0,β). Accordingly,
we further prove that the largest singular value undergoes a phase transition
between the Gaussian and GUE Tracy-Widom distributions.Comment: Therems 1.1 stated in a more intuitive way; proofs extensively
revised; convergence in trace norm for the critical and subcritical cases
added; 35 pages, 3 figure