When does the top homology of a random simplicial complex vanish?1. Issue 1 (13th March 2013) - Aronshtam, Lior; Linial, Nathan
Abstract Several years ago Linial and Meshulam (Combinatorica 26 (2006) 457–487) introduced a model called <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rd4f8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Xd(n, p)</alternatives> of random n‐vertex d‐dimensional simplicial complexes. The following question suggests itself very naturally: What is the threshold probability <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rd4x0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />p=p(n)</alternatives> 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. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 26–35, 2015