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Lower large deviations for geometric functionals in sparse, critical and dense regimes

Christian Hirsch, Daniel Willhalm

TL;DR

This work develops lower large deviation principles for geometric functionals in sparse, critical, and dense regimes where exponential moments may be absent. The authors generalize the sprinkling technique beyond prior limitations, introducing regime-specific couplings (and, in the critical case, dual representations) to enforce bounded stabilization and locality. They establish entropy-based rate functions for sparse and critical regimes, and a tailored dense-regime framework for volume-like kNN functionals, with concrete examples including subgraph counts, Betti numbers, and power-weighted edge lengths. The results provide a unified, technically robust toolkit for analyzing rare events in random geometric graphs and related topological data analysis constructs, with broad potential applications in stochastic geometry and spatial statistics.

Abstract

We prove lower large deviations for geometric functionals in sparse, critical and dense regimes. Our results are tailored for functionals with nonexisting exponential moments, for which standard large deviation theory is not applicable. The primary tool of the proofs is a sprinkling technique that, adapted to the considered functionals, ensures a certain boundedness. This substantially generalizes previous approaches to tackle lower tails with sprinkling. Applications include subgraph counts, persistent Betti numbers and edge lengths based on a sparse random geometric graph, power-weighted edge lengths of a $k$-nearest neighbor graph as well as power-weighted spherical contact distances in a critical regime and volumes of $k$-nearest neighbor balls in a dense regime.

Lower large deviations for geometric functionals in sparse, critical and dense regimes

TL;DR

This work develops lower large deviation principles for geometric functionals in sparse, critical, and dense regimes where exponential moments may be absent. The authors generalize the sprinkling technique beyond prior limitations, introducing regime-specific couplings (and, in the critical case, dual representations) to enforce bounded stabilization and locality. They establish entropy-based rate functions for sparse and critical regimes, and a tailored dense-regime framework for volume-like kNN functionals, with concrete examples including subgraph counts, Betti numbers, and power-weighted edge lengths. The results provide a unified, technically robust toolkit for analyzing rare events in random geometric graphs and related topological data analysis constructs, with broad potential applications in stochastic geometry and spatial statistics.

Abstract

We prove lower large deviations for geometric functionals in sparse, critical and dense regimes. Our results are tailored for functionals with nonexisting exponential moments, for which standard large deviation theory is not applicable. The primary tool of the proofs is a sprinkling technique that, adapted to the considered functionals, ensures a certain boundedness. This substantially generalizes previous approaches to tackle lower tails with sprinkling. Applications include subgraph counts, persistent Betti numbers and edge lengths based on a sparse random geometric graph, power-weighted edge lengths of a -nearest neighbor graph as well as power-weighted spherical contact distances in a critical regime and volumes of -nearest neighbor balls in a dense regime.
Paper Structure (17 sections, 12 theorems, 218 equations, 2 figures)

This paper contains 17 sections, 12 theorems, 218 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Sparse regime
  4. Critical regime
  5. Dense regime
  6. Outline
  7. Examples
  8. Functionals for critical spatial random networks
  9. Power-weighted edge lengths of the directed k-nearest neighbor graph
  10. Power-weighted spherical contact distances
  11. Functionals for the sparse random geometric graph
  12. Subgraph counts
  13. Betti numbers and persistent Betti numbers
  14. Edge lengths
  15. Proof of Theorem \ref{['theorem_main_thermodynamic']} (critical)
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that INV, LOC, BND and POS are satisfied and assume that $k_0\in[1,\infty)$. If $n r_n^d\rightarrow 0$ and $\rho_{n,k_0}^\mathsf{sp}\rightarrow\infty$, then, for $a\in\mathbb R$ and where $T^\mathsf{sp}(\rho) := \int_{(0,\infty)} x {\rm d}\rho(x)$.

Figures (2)

  • Figure 1: Illustrations of a random geometric graph in a sparse, critical and dense regime.
  • Figure 2: Simulation of spherical contact distances based on a Poisson point process on a two-dimensional torus. The lighter the shade, the smaller the distance of a space point to its closest node in the configuration.

Theorems & Definitions (27)

  • Remark 1
  • Theorem 1: Lower large deviations in the sparse regime
  • Remark 2
  • Theorem 2: Lower large deviations in the critical regime
  • Theorem 3: Lower large deviations in the dense regime
  • proof : Proof of Theorem \ref{['theorem_main_thermodynamic']} a)
  • Lemma 4: Sprinkling regularizes with high probability
  • proof : Proof of Lemma \ref{['lemma_sprinkling']}
  • proof : Proof of Theorem \ref{['theorem_main_thermodynamic']} b)
  • Lemma 5: Bad boxes are exponentially negligible
  • ...and 17 more