TY - JOUR
T1 - The homology of random simplicial complexes in the multi-parameter upper model
AU - Farber, Michael
AU - Nowik, Tahl
N1 - Publisher Copyright:
© 2023, The Hebrew University of Jerusalem.
PY - 2023/9
Y1 - 2023/9
N2 - 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 homology in the critical dimension ℓ is significantly the largest, and we specify the precise rate of growth of the ℓth Betti number. For the remaining Betti numbers in this range we give upper bounds that strongly decrease from dimension to dimension.
AB - 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 homology in the critical dimension ℓ is significantly the largest, and we specify the precise rate of growth of the ℓth Betti number. For the remaining Betti numbers in this range we give upper bounds that strongly decrease from dimension to dimension.
UR - http://www.scopus.com/inward/record.url?scp=85173689973&partnerID=8YFLogxK
U2 - 10.1007/s11856-023-2507-7
DO - 10.1007/s11856-023-2507-7
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AN - SCOPUS:85173689973
SN - 0021-2172
VL - 256
SP - 213
EP - 231
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -