On the phase transition in random simplicial complexes
نویسنده
چکیده
It is well known that the G(n, p) model of random graphs undergoes a dramatic change around p = 1 n . It is here that the random graph, almost surely, contains cycles, and here it first acquires a giant (i.e., order Ω(n)) connected component. Several years ago, Linial and Meshulam introduced the Yd(n, p) model, a probability space of n-vertex d-dimensional simplicial complexes, where Y1(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 Yd(n, p). We also compute the real Betti numbers of Yd(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 ≥ 2 the emergence of the giant shadow is a first order phase transition.
منابع مشابه
Random Simplicial Complexes - Around the Phase Transition
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