Let ∆n−1 denote the (n − 1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of ∆n−1 obtained by starting with the full 1-dimensional skeleton of ∆n−1 and then adding each 2−simplex independently with probability p. Let H1(Y ; F2) denote the first homology group of Y with mod 2 coefficients. It is shown that for any function ω(n) that tends to infinity lim n→∞ Prob[H1(Y ; F2) = 0] = 0 p = 2 log n−ω(n) n 1 p = 2 log n+ω(n) n .
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 H k−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
Let ∆ n−1 denote the (n − 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of ∆ n−1 obtained by starting with the full (d − 1)-dimensional skeleton of ∆ n−1 and then adding each d-simplex independently with probability p = c n . We compute an explicit constant γ d = Θ(log d) so that for c < γ d such a random simplicial complex either collapses to a (d−1)-dimensional subcomplex or it contains ∂∆ d+1 , the boundary of a (d + 1)-dimensional simplex. We conjecture this bound to be sharp. In addition we show that there exists a constant γ d < c d < d + 1 such that for any c > c d and a fixed field F, asymptotically almost surely H d (Y ; F) = 0.
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