Large deviation principle for geometric and topological functionals and associated point processes

TL;DR

This paper establishes a large deviation principle for point processes related to connected components in a Poisson point process, with applications to topological invariants like Betti numbers and Morse critical points.

Contribution

It introduces a large deviation principle for geometric and topological functionals in a sparse regime, with a rate function expressed as relative entropy, advancing stochastic geometry and topology analysis.

Findings

Large deviation principle for connected component point processes.

Rate function expressed as relative entropy.

Applications to Betti numbers and Morse critical points.

Abstract

We prove a large deviation principle for the point process associated to $k$-element connected components in $\mathbb R^d$ with respect to the connectivity radii $r_n\to\infty$. The random points are generated from a homogeneous Poisson point process, so that $(r_n)_{n\ge1}$ satisfies $n^kr_n^{d(k-1)}\to\infty$ and $nr_n^d\to0$ as $n\to\infty$ (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.