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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
Zbl 1121.55013
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