Abstract
In this paper we study the spectrum of the random geometric graph , in a regime where the graph is dense and highly connected. In the Erdős–Rényi random graph it is well known that upon connectivity the spectrum of the normalized graph Laplacian is concentrated around 1. We show that such concentration does not occur in the case, even when the graph is dense and almost a complete graph. In particular, we show that the limiting spectral gap is strictly smaller than 1. In the special case where the vertices are distributed uniformly in the unit cube and , we show that for every there are at least eigenvalues near , and the limiting spectral gap is exactly . We also show that the corresponding eigenfunctions in this case are tightly related to the geometric configuration of the points.
Funding Statement
The research of KA was supported in part by the Zeff Fellowship, Viterbi Fellowship and the Israel Science Foundation, Grants 2539/17, 771/17, and 1965/19. The research of RJA was supported in part by the Israel Science Foundation, Grant 2539/17. The research of OB was supported in part by by the Israel Science Foundation, Grant 1965/19. The research of RR was supported in part by Israel Science Foundation, Grant 771/17 and Binational Science Foundation, Grant 2029528.
Citation
Kartick Adhikari. Robert J. Adler. Omer Bobrowski. Ron Rosenthal. "On the spectrum of dense random geometric graphs." Ann. Appl. Probab. 32 (3) 1734 - 1773, June 2022. https://doi.org/10.1214/21-AAP1720
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