Random moment angle complex

Abstract

Let n be a positive integer and p ∈ [0, 1]. The random simplicial d-complex Y d n;p on n vertices and with parameter p, introduced by Meshulam and Wallach, has the following probability law:- Y d n;p takes values in the space of all simplicial complexes K such that ∆n d−1 ⊂ K ⊂ ∆n d - each possible d-simplex of ∆n d appears in Y d n;p with probability p, independently. We introduce moment angle complex over the random simplicial complex and study its topological and combinatorial features. Specially, some Law of the large numbers for the bigraded Betti numbers has been established. Several other directions for studying randomness in toric topology have been considered.