The fundamental group of the 2-dimensional Linial–Meshulam random simplicial complex Y2(n, p) was first studied by Babson, Hoffman, and Kahle. They proved that the threshold probability for simple connectivity of Y2(n, p) is about p≈ n- 1 / 2. In this paper, we show that this threshold probability is at most p≤ (γn) - 1 / 2, where γ= 4 4/ 3 3, and conjecture that this threshold is sharp. In fact, we show that p= (γn) - 1 / 2 is a sharp threshold probability for the stronger property that every cycle of length 3 is the boundary of a subcomplex of Y2(n, p) that is homeomorphic to a disk. Our proof uses the Poisson paradigm, and relies on a classical result of Tutte on the enumeration of planar triangulations.
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- Poisson paradigm
- Simple connectivity
- Simplicial complexes
- Threshold probability