  We study random transformations built from intermittent maps on the unit
interval that share a common neutral fixed point. We focus mainly on random
selections of Pomeu-Manneville-type maps $T_\alpha$ using the full parameter
range $0< \alpha < \infty$, in general. We derive a number of results around a
common theme that illustrates in detail how the constituent map that is fastest
mixing (i.e.\ smallest $\alpha$) combined with details of the randomizing
process, determines the asymptotic properties of the random transformation. Our
key result (Theorem 1.1) establishes sharp estimates on the position of return
time intervals for the \emph{quenched} dynamics. The main applications of this
estimate are to \textit{limit laws} (in particular, CLT and stable laws,
depending on the parameters chosen in the range $0 < \alpha < 1$) for the
associated skew product; these are detailed in Theorem 3.2. Since our estimates
in Theorem 1.1 also hold for $1 \leq \alpha < \infty$ we study a piecewise
affine version of our random transformations, prove existence of an infinite
($\sigma-$finite) invariant measure and study the corresponding correlation
asymptotics. To the best of our knowledge, this latter kind of result is
completely new in the setting of random transformations.
