  For a family of random intermittent dynamical systems with a superattracting
fixed point we prove that a phase transition occurs between the existence of an
absolutely continuous invariant probability measure and infinite measure
depending on the randomness parameters and the orders of the maps at the
superattracting fixed point. In case the systems have an absolutely continuous
invariant probability measure, we show that the systems are mixing and that the
correlations decay polynomially even though some of the deterministic maps
present in the system have exponential decay. This contrasts other known
results, where random systems adopt the best decay rate of the deterministic
maps in the systems.
