  We derive exact analytical expressions for correlation functions of singular
values of the product of $M$ Ginibre matrices of size $N$ in the double scaling
limit $M,N\rightarrow \infty$. The singular value statistics is described by a
determinantal point process with a kernel that interpolates between GUE
statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit,
$N\rightarrow \infty$, the interpolation parameter is given by the limiting
quotient $a=N/M$. One of our goals is to find an explicit form of the kernel at
the hard edge, in the bulk and at the soft edge for any $a$. We find that in
addition to the standard scaling regimes, there is a new transitional regime
which interpolates between the hard edge and the bulk. We conjecture that these
results are universal, and that they apply to a broad class of products of
random matrices from the Gaussian basin of attraction, including correlated
matrices. We corroborate this conjecture by numerical simulations.
Additionally, we show that the local spectral statistics of the considered
random matrix products is identical with the local statistics of Dyson Brownian
motion with the initial condition given by equidistant {positions,} with the
crucial difference that this equivalence holds only locally. Finally, we have
identified a mesoscopic spectral scale at the soft edge which is crucial for
the unfolding of the spectrum.
