  We present a null model for single- and multi-layered complex systems
constructed using homogeneous and isotropic random Gaussian maps. By means of a
Kac-Rice formalism, we show that the mean number of fixed points can be
calculated as the expectation of the absolute value of the characteristic
polynomial for a product of independent Gaussian (Ginibre) matrices.
Furthermore, using techniques from Random Matrix Theory, we show that the
high-dimensional limit of our system has a third-order phase transition between
a phase with a single fixed point and a phase with exponentially many fixed
points. This is result is universal in the sense that it does not depend on
finer details of the correlations for the random maps.
