  We describe topology of random simplicial complexes in the lower and upper
models in the medial regime, i.e. under the assumption that the probability
parameters $p_\sigma$ approach neither $0$ nor $1$. We show that nontrivial
Betti numbers of typical lower and upper random simplicial complexes in the
medial regime lie in a narrow range of dimensions. For instance, an upper
random simplicial complex $Y$ on $n$ vertices in the medial regime with high
probability has non-vanishing Betti numbers $b_{j}(Y)$ only for $k+c
<n-j<k+\log_2 k +c'$ where $k=\log_2 \ln n$ and $c, c' $ are constants. A lower
random simplicial complex on $n$ vertices in the medial regime is with high
probability $(k+a)$-connected and its dimension $d$ satisfies $d\sim k+\log_2
k+ a'$ where $a, \, a'$ are constants. The paper develops a new technique,
based on Alexander duality, which relates the lower and upper models.
