Erratum to “Limit theorems for Betti numbers of random simplicial complexes”
Matthew Kahle, Elizabeth Meckes · Homology Homotopy and Applications · 2016
We correct the proofs of the main theorems in our earlier paper "Limit theorems for Betti numbers of random simplicial complexes."We are grateful to D. Yogeshwaran for pointing out the mistakes.Theorem 1.1.Consider the Erdős-Renyi clique complex X(n, p); that is, take a random 1-skeleton on n vertices, in which edges are present independently and with probability p, and let X(n, p) be the maximal complex over this 1-skeleton.Suppose that there is some δ > 0 such that p = ω(n -1/k+δ ) and p = o(n -1/(k+1)-δ ).Then⇒ N(0, 1).Note that the range of p is slightly restricted relative to what was claimed earlier, when δ was taken to be 0.The mistake in the proof given in [2] of this result was the claim that, given the Morse inequalitiesand central limit theorems for the recentered, renormalized upper and lower bounds, a central limit theorem for β k itself followed; this is not true, however, because the M. Kahle's research was supported in part by Stanford's NSF-RTG in geometry and topology.