On topological minors in random simplicial complexes

Abstract

For random graphs, the containment problem considers the probability that a binomial random graph G(n, p) contains a given graph as a substructure. When asking for the graph as a topological minor, i.e. for a copy of a subdivision of the given graph, it is well-known that the (sharp) threshold is at p = 1/n. We consider a natural analogue of this question for higher-dimensional random complexes X k (n, p), first studied by Cohen, Costa, Farber and Kappeler for k = 2.Improving previous results, we show that p = Θ(1/ √ n) is the (coarse) threshold for containing a subdivision of any fixed complete 2-complex. For higher dimensions k > 2, we get that p = O(n −1/k ) is an upper bound for the threshold probability of containing a subdivision of a fixed k-dimensional complex.

Integral Homology of Random Simplicial Complexes

“…(1) Denote by b i (C) the number of bad i-dimensional faces for i = 0, 1. Then, b 1 (C) ≤ 3n 2 ln (5) n ln ln n , b 0 (C) ≤ 6n ln (5) n · ln (4) n ln ln n ≤ n ln (4) n .…”

“…Proof of Lemma 6. Let Y = Y 2 (n, p), where np = 2 ln n−ln (4) n. The proof of Lemma 6 is done in three steps. Initially, in Corollary 10 we show that a.a.s in every Y -shady partition, there is a big subset W of vertices, such that all the triangles of W are good.…”

“…(2) Suppose that x and y are good vertices. Then, there exists at least n − 3n ln (4) n good vertices v such that the edges vx and vy are good.…”

“…We start by selecting the set of good vertices and m bad edges on these good vertices, for m ≥ m 0 = n/ ln ln n. Afterwards, we sample the complex Y and bound the failure probability that the selected configuration can be induced by a Y -shady partition. On the one hand, our selection must be consistent with the second item of Claim 8, thus for every bad edge xy there are at least n − 3n ln (4) n vertices v such that both xv and yv are good. On the other hand, by the third item of the claim, the triangle vxy must be bad for every such xy and v. Therefore, our selection points at at least mn(1 − o(1)) distinct triangles that Y does not contain.…”

“…Babson, Hoffman and Kahle proved in [1] that the threshold for simple-connectivity is of order n − 1 2 , substantially larger than that for homological connectivity. Tighter upper bounds were obtained in [10,4] The Linial-Meshulam theorem was also generalized in the direction of a hitting-time result. Consider the random 2-dimensional complex process Y 2 = {Y 2 (n, M) : 0 ≤ M ≤ n 3 }, defined as the Markov chain in which we start with a complete graph on n vertices, and in every step M a new 2-dimensional face, chosen uniformly at random, is added.…”