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Limit theorems for high-dimensional Betti numbers in the multiparameter random simplicial complexes

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  • Owada, Takashi
  • Samorodnitsky, Gennady

Abstract

We consider the multiparameter random simplicial complex on a vertex set {1,…,n}, which is parameterized by multiple connectivity probabilities. Our key results concern the topology of this complex of dimensions higher than the critical dimension. We show that the higher-dimensional Betti numbers satisfy strong laws of large numbers and central limit theorems. Moreover, lower tail large deviations for these Betti numbers are also discussed. Some of our results indicate an occurrence of phase transitions in terms of the scaling constants of the central limit theorem, and the exponentially decaying rate of convergence of lower tail large deviation probabilities.

Suggested Citation

  • Owada, Takashi & Samorodnitsky, Gennady, 2025. "Limit theorems for high-dimensional Betti numbers in the multiparameter random simplicial complexes," Stochastic Processes and their Applications, Elsevier, vol. 186(C).
    • Handle: RePEc:eee:spapps:v:186:y:2025:i:c:s0304414925000821
    DOI: 10.1016/j.spa.2025.104641
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    1. Owada, Takashi & Samorodnitsky, Gennady & Thoppe, Gugan, 2021. "Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 56-95.
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