The random $d$-complex $Y_d (n, p)$ has the vertex set $[n]$ contains, the complete $(d-1)$-skeleton of a simplex on $n$ vertices and each possible $d$-dimensional face appears independently with probability $p$. It is a generalization of

Erd\"{o}s-R\'{e}nyi model of the random graph. Using constructions in toric spaces we assign to $Y_d (n, p)$ certain `random toric spaces' and study their topological and geometrical properties. Particularly, we establish the law of large numbers for the bigraded Betti numbers of $Y_d (n, p)$.

This is a joint work with Vlada Limi\'{c}.