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Central limit theorem for linear eigenvalue statistics of random geometric graphs

Christian Hirsch, Kyeongsik Nam, Moritz Otto

Abstract

Random spatial networks-that is, graphs whose connectivity is governed by geometric proximity-have emerged as fundamental models for systems constrained by an underlying spatial structure. A prototypical example is the random geometric graph, obtained by placing vertices according to a Poisson point process and connecting two vertices whenever their Euclidean distance is less than a certain threshold. Despite their broad applicability, the spectral properties of such spatial models remain far less understood than those of classical random graph models, such as Erdős-Rényi graphs and Wigner matrices. The main obstacle is the presence of spatial constraints, which induce highly nontrivial dependencies among edges, placing these models outside the scope of techniques developed for purely combinatorial random graphs. In this paper, we provide the first rigorous analysis of Gaussian fluctuations for linear eigenvalue statistics of random geometric graphs. Specifically, we establish central limit theorems for $\text{Tr}[φ(A)]$, where $A$ is the adjacency matrix and $φ$ ranges over a broad class of suitable (possibly non-polynomial) test functions. In the polynomial setting, we moreover obtain a quantitative central limit theorem, including an explicit convergence rate to the limiting Gaussian law. We further obtain polynomial-test-function CLTs for other canonical random spatial networks, including $k$-nearest neighbor graphs and relative neighborhood graphs. Our results open new avenues for the study of spectral fluctuations in spatially embedded random structures and underscore the delicate interplay between geometry, local dependence, and spectral behavior.

Central limit theorem for linear eigenvalue statistics of random geometric graphs

Abstract

Random spatial networks-that is, graphs whose connectivity is governed by geometric proximity-have emerged as fundamental models for systems constrained by an underlying spatial structure. A prototypical example is the random geometric graph, obtained by placing vertices according to a Poisson point process and connecting two vertices whenever their Euclidean distance is less than a certain threshold. Despite their broad applicability, the spectral properties of such spatial models remain far less understood than those of classical random graph models, such as Erdős-Rényi graphs and Wigner matrices. The main obstacle is the presence of spatial constraints, which induce highly nontrivial dependencies among edges, placing these models outside the scope of techniques developed for purely combinatorial random graphs. In this paper, we provide the first rigorous analysis of Gaussian fluctuations for linear eigenvalue statistics of random geometric graphs. Specifically, we establish central limit theorems for , where is the adjacency matrix and ranges over a broad class of suitable (possibly non-polynomial) test functions. In the polynomial setting, we moreover obtain a quantitative central limit theorem, including an explicit convergence rate to the limiting Gaussian law. We further obtain polynomial-test-function CLTs for other canonical random spatial networks, including -nearest neighbor graphs and relative neighborhood graphs. Our results open new avenues for the study of spectral fluctuations in spatially embedded random structures and underscore the delicate interplay between geometry, local dependence, and spectral behavior.
Paper Structure (26 sections, 13 theorems, 141 equations, 2 figures)

This paper contains 26 sections, 13 theorems, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and main results
  3. General test functions
  4. Polynomials: Quantitative convergence rates
  5. Related works: Erdős–Rényi graphs and random regular graphs
  6. Erdős–Rényi graph
  7. Random $d$-regular graph
  8. Organization
  9. Acknowledgement
  10. Preliminaries on the Poisson point process
  11. Mecke formula
  12. Difference operator
  13. Poincaré inequality
  14. Proof of Theorem \ref{['thm:polt1']}--Polynomials
  15. Variance asymptotics
  16. ...and 11 more sections

Key Result

Theorem 1

Fix $r>0$, and let $A_n$ denote the adjacency matrix of $\mathrm{RGG}(r)$ on the vertex set $\mathcal{P}_n$. Let $f: \mathbb R \rightarrow \mathbb R$ be a twice weakly differentiable function with $f(0)=0$ such that for some constant $c\neq 0$, where $\textup{sech} (y) := \frac{2}{e^{y} + e^{-y}}$ denotes the reciprocal of $\cosh (y):= \frac{e^y+e^{-y}}{2}$. Then the central limit theorem holds:

Figures (2)

  • Figure 1: Illustration of the stabilization radius in the RNG
  • Figure 2: The Apollonius circle of all points $z$ with equal ratio $\frac{|x-z|}{|X_i-z|}=\frac{|x-(X_i+X_j)/2|}{|X_i-(X_i+X_j)/2|}$ is drawn in blue. In particular, this ratio is larger for all points inside this circle.

Theorems & Definitions (24)

  • Theorem 1: CLT for linear eigenvalue statistics; general test functions
  • Theorem 2: CLT for linear eigenvalue statistics; polynomials
  • Lemma 3.1: Mecke formula, Theorem 4.1 in poisBook
  • Lemma 3.2: Multi-dimensional Mecke formula, Theorem 4.4 in poisBook
  • Lemma 3.3: Poincaré inequality
  • proof : Proof of the variance asymptotics
  • proof : Proof of the variance positivity
  • Lemma 4.1: Fourth moment bounds for the difference operators
  • proof : Proof of Theorem \ref{['thm:polt1']}
  • Lemma 4.2: Moment bounds
  • ...and 14 more