Quantitative results on spectral problems in homogenization

Abstract

It is a classical result on periodic homogenization due to S. Kesavan from the late 70's, that the every Dirichlet eigenvalues of a periodic, uniformly elliptic, symmetric operator on a bounded domain converges to the associated Dirichlet eigenvalue of the homogenized operator. In the first part of this talk, we obtain optimal convergence rates for the eigenvalues and eigenfunctions of such operators on all of space, together with a smooth confining potential that grows quadratically at infinity. In the second part of the talk, motivated by some problems in unravelling the geometry of data, we consider similar questions for the spectrum of the graph laplacian, for a graph formed by connecting points with unit distance of each other in a Poisson point process on $R^d.$ When the intensity of the process is greater than the "percolation intensity" which ensures that the graph has a unique unbounded connected component, we show that one can obtain convergence rates for the spectrum that are the same as those in periodic homogenization. This talk is a joint work with Scott N. Armstrong (Courant).