2022/02/16 by Gennady Samorodnitsky, Samorodnitsky, Gennady, Takashi Owada +1
Computer Science · Mathematics · #Data Management and Algorithms #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2202.08330
openalex publication_date 2022/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erdös-Rényi graph. We investigate appearance of "unusual" topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices at the critical dimension and below. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.