Homological connectivity of random k ‐dimensional complexes

Let Δ n −1 denote the ( n − 1)‐dimensional simplex. Let Y be a random k ‐dimensional subcomplex of Δ n −1 obtained by starting with the full ( k − 1)‐dimensional skeleton of Δ 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$${\mathop{\rm lim}\limits_{n \rightarrow \infty}}{\Pr}[H_{k-1} \; (Y;R)= 0]=\left\{ \eqalign {&{0 \quad p= {k\; {\rm log}\;n-\omega(n) \over n }} \cr & 1 \quad p={k\; {\rm log}\; n + \omega (n) \over n}. } \right. $$ © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009