Spectral convergence of graph Laplacians with Ricci curvature bounds and in non-collapsed Ricci limit spaces

Abstract

This paper establishes quantitative high-probability bounds on the eigenvalues and eigenfunctions of $\epsilon$-neighborhood graph Laplacians constructed from i.i.d. random variables on $m$-dimensional closed Riemannian manifolds $(M,g)$ that satisfy a uniform lower Ricci curvature bound ${\mathrm{Ric}}_g\ge -(m-1)K$, a positive lower volume bound, and an upper diameter bound. These results extend to non-collapsed Ricci limit spaces that are measured Gromov-Hausdorff limits of such manifolds, and the bounds give a spectral approximation of weighted Laplacians on manifolds with non-smooth points.

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