Central limit theorem for linear eigenvalue statistics of random geometric graphs

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.

💡 Research Summary

The paper tackles a long‑standing gap in the spectral theory of spatial random graphs by establishing central limit theorems (CLTs) for linear eigenvalue statistics of random geometric graphs (RGGs). An RGG is generated by placing vertices according to a homogeneous Poisson point process in a growing window and connecting any two vertices whose Euclidean distance does not exceed a fixed radius r. The adjacency matrix Aₙ has random size N=|Pₙ|, and the object of interest is the trace functional
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