LARGE DEVIATION PRINCIPLE FOR GEOMETRIC AND TOPOLOGICAL FUNCTIONALS AND ASSOCIATED POINT PROCESSES

• 成果类型: Article
• 署名作者: Hirsch, Christian; Owada, Takashi
• 署名单位: Aarhus University; Purdue University System; Purdue University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1914
• 发表日期: 2023
• 页码: 4008-4043
• 关键词: U-STATISTICS; LIMIT-THEOREMS; distance

摘要

We prove a large deviation principle for the point process associated to k-element connected components in R-d with respect to the connectivity radii r(n)->infinity. The random points are generated from a homogeneous Poisson point process or the corresponding binomial point process, so that (r(n))(n >= 1) satisfies n(k)r(n)(d(k-1))n ->infinity and nr(n)(d) -> 0 as n -> infinity (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.