For Gaussian-sampled random geometric graphs in the sparse regime, the first nontrivial edge eigenvalues of the scaled random-walk Laplacian converge in probability to 2(k-1)/sigma^2-type eigenvalues of a weighted Laplace-Beltrami operator.
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On the edge eigenvalues of sparse random geometric graphs
For Gaussian-sampled random geometric graphs in the sparse regime, the first nontrivial edge eigenvalues of the scaled random-walk Laplacian converge in probability to 2(k-1)/sigma^2-type eigenvalues of a weighted Laplace-Beltrami operator.