Homological connectivity of random k-dimensional complexes

Let ∆n−1 denote the (n − 1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of ∆n−1 obtained by starting with the full (k − 1)-dimensional skeleton of ∆n−1 and then adding each k-simplex independently with probability p. Let Hk−1(Y ;R) denote the (k−1)-dimensional reduced homology group of Y with coefficients in a finite abelian group R. It is shown that for any fixed R and k ≥ 1 and for any function ω(n) that tends to infinity lim n→∞ Pr [ Hk−1(Y ;R) = 0 ] = { 0 p = k logn−ω(n) n 1 p = k logn+ω(n) n