  A square-tiled surface (STS) is a branched cover of the standard square torus
with branching over exactly one point. In this paper we consider a randomizing
model for STSs and generalizations to branched covers of other simple
translation surfaces which we call polygon-tiled surfaces. We obtain a local
central limit theorem for the genus and subsequently obtain that the
distribution of the genus is asymptotically normal. We also study holonomy
vectors (Euclidean displacement vectors between cone points) on a random STS.
We show that asymptotically almost surely the set of holonomy vectors of a
random STS contains the set of primitive vectors of $\mathbb{Z}^2$ and with
probability approaching $1/e$, these sets are equal.
