On the phase transition in random simplicial complexes

It is well known that the G(n,p) model of random graphs undergoes a dramatic change around p = 1/n . It is here that the random graph, almost surely, contains cycles, and here it first acquires a giant (i.e., order ω(n))connected component. Several years ago, Linial and Meshulam introduced the Y_d(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 natural d-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 Y_d(n,p). We also compute the real Betti numbers of Y_d(n,p) for p = 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.