Lower large deviations for geometric functionals

TL;DR

This paper introduces a new methodology for analyzing the lower tail large deviations of geometric functionals on Poisson point processes, with applications to various geometric graph characteristics.

Contribution

It develops a general approach for lower tail large deviations of stabilizing geometric functionals, applicable to multiple types of geometric graphs and structures.

Findings

Derived bounds for clique counts in random geometric graphs

Analyzed intrinsic volumes of Poisson-Voronoi cells

Studied power-weighted edge lengths in various geometric graphs

Abstract

This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson-Voronoi cells, as well as power-weighted edge lengths in the random geometric, $k$-nearest neighbor and relative neighborhood graph.