2022/09/12 by Farber, Michael, Nowik, Tahl
#Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2209.05418
We study random simplicial complexes in the multi-parameter upper model. In this model simplices of various dimensions are taken randomly and independently, and our random simplicial complex Y is then taken to be the minimal simplicial complex containing this collection of simplices. We study the asymptotic behavior of the homology of Y as the number of vertices goes to ∞. We observe the following phenomenon asymptotically almost surely. The given probabilities with which the simplices are taken determine a range of dimensions ℓ ≤ k ≤ ℓ' with ℓ' ≤ 2ℓ +1, outside of which the homology of Y vanishes. Within this range, the homologies diminish drastically from dimension to dimension. In particular, the homology in the critical dimension ℓ is significantly the largest.