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On the Dimension of Random Simplicial Complexes

Kinga Nagy

TL;DR

This work analyzes the extreme-dimensional functional of random geometric simplicial complexes built on Poisson point processes, linking dimension to both clique numbers and scan statistics. It develops a unified, Poisson-based framework to obtain expectations, large deviations principles, two-point concentration in sparse regimes, and precise distribution results in the dense regime for both Vietoris-Rips and Čech complexes. The results reveal regime-dependent behavior: sparse settings exhibit sharp two-point concentration while dense regimes exhibit Poisson-type distributional limits, with a special Gumbel limit in 1D. Overall, the paper advances understanding of extreme-valued functionals on random geometric complexes and provides tools for related scan-statistics analyses.

Abstract

The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique number of geometric graphs or scan statistics. We extend existing results in the binomial point process case to the Poisson setting in sparse graphs, give new ones about expectations and large deviation principles in all regimes, as well as give a first precise distribution result in the dense case.

On the Dimension of Random Simplicial Complexes

TL;DR

This work analyzes the extreme-dimensional functional of random geometric simplicial complexes built on Poisson point processes, linking dimension to both clique numbers and scan statistics. It develops a unified, Poisson-based framework to obtain expectations, large deviations principles, two-point concentration in sparse regimes, and precise distribution results in the dense regime for both Vietoris-Rips and Čech complexes. The results reveal regime-dependent behavior: sparse settings exhibit sharp two-point concentration while dense regimes exhibit Poisson-type distributional limits, with a special Gumbel limit in 1D. Overall, the paper advances understanding of extreme-valued functionals on random geometric complexes and provides tools for related scan-statistics analyses.

Abstract

The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique number of geometric graphs or scan statistics. We extend existing results in the binomial point process case to the Poisson setting in sparse graphs, give new ones about expectations and large deviation principles in all regimes, as well as give a first precise distribution result in the dense case.

Paper Structure

This paper contains 5 sections, 8 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the expectation and LDP
  4. Proof of the two-point concentration
  5. The dense regime

Key Result

Theorem 1

Let $\eta_t$ be a homogeneous Poisson point process on the unit volume compact convex body $W\subset \mathbb{R}^d$ with intensity $t>0$. Choose $r_t>0$ such that $r_t\to 0$ as $t \to \infty$, and let $\rho_t:=tr_t^d$. Let $D$ denote the dimension of the Vietoris-Rips or Čech complexes built on $\eta

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Corollary
  • ...and 1 more