L. Aronshtam

Publications (2)0 Total impact

• Article: When does the top homology of a random simplicial complex vanish?

  L. Aronshtam, N. Linial
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  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.
  03/2012;
• Source

  Available from: ArXiv

  Article: Collapsibility and vanishing of top homology in random simplicial complexes

  L. Aronshtam, N. Linial, T. Luczak, R. Meshulam
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  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 0.
  10/2010;