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The spectrum of a random geometric graph is concentrated

Sanatan Rai
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Abstract

Consider $n$ points distributed uniformly in $[0,1]^d$. Form a graph by connecting two points if their mutual distance is no greater than $r(n)$. This gives a random geometric graph, $\gnrn$, which is connected for appropriate $r(n)$. We show that the spectral measure of the transition matrix of the simple random walk (\abbr{srw}) on $\gnrn$ is concentrated, and in fact converges to that of the graph on the deterministic grid.

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Sanatan Rai. 2004-09-23. The spectrum of a random geometric graph is concentrated. https://arxiv.org/abs/math/0408103

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