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On the phase transition in random simplicial complexes. (English) Zbl 1348.05193
Summary: It is well known that the \(G(n,p)\) model of random graphs undergoes a dramatic change around \(p=\frac{1}{n}\). It is here that the random graph, almost surely, contains cycles, and here it first acquires a giant (i.e., order \(\Omega(n))\) connected component. Several years ago, N. Linial and R. Meshulam [Combinatorica 26, No. 4, 475–487 (2006; Zbl 1121.55013)] introduced the \(Y_d(n,p)\) model, a probability space of \(n\)-vertex \(d\)-dimensional simplicial complexes, where \(Y_1(n,p)\) coincides with \(G(n,p)\). Within this model we prove a natural \(d\)-dimensional analog of these graph theoretic phenomena. Specifically, we determine the exact threshold for the nonvanishing of the real \(d\)-th homology of complexes from \(Y_d(n,p)\). We also compute the real Betti numbers of \(Y_d(n,p)\) for \(p=c/n\). Finally, we establish the emergence of giant shadow at this threshold. (For \(d=1\), a giant shadow and a giant component are equivalent). Unlike the case for graphs, for \(d\geq 2\) the emergence of the giant shadow is a first order phase transition.

MSC:
05C80 Random graphs (graph-theoretic aspects)
05E45 Combinatorial aspects of simplicial complexes
60C05 Combinatorial probability
Citations:
Zbl 1121.55013
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