On the phase transition in random simplicial complexes

Abstract

It is well-known that the G(n;p) model of random graphs undergoes a dramatic change around p = 1 . It is here that the random graph, almost surely, contains cycles, and here it first acquires a giant (i.e., order ( n)) connected component. Several years ago, Linial and Meshulam have introduced theYd(n;p) model, a probability space of n-vertex d-dimensional simplicial complexes, where Y1(n;p) coincides with G(n;p). Within this model we prove a naturald-dimensional analog of these graph theoretic phenomena. Specifically, we determine the exact threshold for the nonvanishing of the real d-th homology of complexes from Yd(n;p). We also compute the real Betti numbers ofYd(n;p) forp = c=n. Finally, we establish the emergence of giant shadow at this threshold. (For d = 1 a giant shadow and a giant component are equivalent). Unlike the case for graphs, for d 2 the emergence of the giant shadow is a first order phase transition.