  Let $\mathcal{C}(n,k)$ be the set of $k$-dimensional simplicial complexes $C$
over a fixed set of $n$ vertices such that:
  (1) $C$ has a complete $k-1$-skeleton;
  (2) $C$ has precisely ${{n-1}\choose {k}}$ $k$-faces;
  (3) the homology group $H_{k-1}(C)$ is finite.
  Consider the probability measure on $\mathcal{C}(n,k)$ where the probability
of a simplicial complex $C$ is proportional to $|H_{k-1}(C)|^2$. For any fixed
$k$, we determine the local weak limit of these random simplicial complexes as
$n$ tends to infinity.
  This local weak limit turns out to be the same as the local weak limit of the
$1$-out $k$-complexes investigated by Linial and Peled.
