The random simplicial complex and toric topology

Abstract

The random d-complex Yd(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 Yd(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 Yd(n,p).