  We consider Random Matrix Theories with non-Gaussian potentials that have a
rich phase structure in the large $N$ limit. We calculate the Spectral Form
Factor (SFF) in such models and present them as interesting examples of
dynamical models that display multi-criticality at short time-scales and
universality at large time scales. The models with quartic and sextic
potentials are explicitly worked out. The disconnected part of the Spectral
Form Factor (SFF) shows a change in its decay behavior exactly at the critical
points of each model. The dip-time of the SFF is estimated in each of these
models. The late time behavior of all polynomial potential matrix models is
shown to display a certain universality. This is related to the universality in
the short distance correlations of the mean-level densities. We speculate on
the implications of such universality for chaotic quantum systems including the
SYK model.
