The spectrum of a random geometric graph is concentrated

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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,, which is connected for appropriate r(n). We show that the spectral measure of the transition matrix of the simple random walk (srw) on is concentrated, and in fact converges to that of the graph on the deterministic grid.