Random Simplicial Complexes in the Medial Regime

2019/07/01 by Farber, Michael, Mead, Lewis
#Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1907.00653

Abstract

We describe topology of random simplicial complexes in the lower and upper models in the medial regime, i.e. under the assumption that the probability parameters p_σ approach neither 0 nor 1. We show that nontrivial Betti numbers of typical lower and upper random simplicial complexes in the medial regime lie in a narrow range of dimensions. For instance, an upper random simplicial complex Y on n vertices in the medial regime with high probability has non-vanishing Betti numbers b_j(Y) only for k+c

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