Publications (2)0 Total impact
ABSTRACT: Several years ago Linial and Meshulam introduced a model called X_d(n,p) of
random n-vertex d-dimensional simplicial complexes. The following question
suggests itself very naturally: What is the threshold probability p=p(n) at
which the d-dimensional homology of such a random d-complex is, almost surely,
nonzero? Here we derive an upper bound on this threshold. Computer experiments
that we have conducted suggest that this bound may coincide with the actual
threshold, but this remains an open question.
ABSTRACT: Let Y be a random d-dimensional subcomplex of the (n-1)-dimensional simplex S
obtained by starting with the full (d-1)-dimensional skeleton of S and then
adding each d-simplex independently with probability p=c/n. We compute an
explicit constant gamma_d=Theta(log d) so that for c < gamma_d such a random
simplicial complex either collapses to a (d-1)-dimensional subcomplex or it
contains the boundary of a (d+1)-simplex. We conjecture this bound to be sharp.
In addition we show that there exists a constant gamma_d< c_d <d+1 such that
for any c>c_d and a fixed field F, asymptotically almost surely H_d(Y;F) \neq