  In this paper we determine the threshold for collapsibility in the
probabilistic model $X_d(n,p)$ of $d$-dimensional simplicial complexes. A lower
bound for this threshold $p=\frac{c_d}{n}$ was established in \cite{ALLM}. Here
we show that this is indeed the correct threshold. Namely, for every $c>c_d$, a
complex drawn from $X_d(n,\frac{c}{n})$ is asymptotically almost surely not
collapsible.
