Eigenvalue fluctuations for random elliptic operators in homogenization regime

TL;DR

This paper investigates the asymptotic behavior and fluctuations of eigenvalues of elliptic operators with random coefficients in homogenization, providing convergence rates, two-scale expansions, and a central limit theorem.

Contribution

It introduces a quantitative analysis of eigenvalue fluctuations, linking them to the homogenization commutator, and advances the understanding of spectral properties in random homogenization.

Findings

Sharp convergence rates for eigenvalues to homogenized eigenvalues

Quantitative two-scale expansion for eigenfunctions

Pathwise characterization of eigenvalue fluctuations via homogenization commutator

Abstract

This work is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is obtained for isolated eigenvalues towards eigenvalues of the homogenized problem, as well as a quantitative two-scale expansion result for eigenfunctions. Next, a quantitative central limit theorem is established for eigenvalue fluctuations; more precisely, a pathwise characterization of eigenvalue fluctuations is obtained in terms of the so-called homogenization commutator, in parallel with the recent fluctuation theory for the solution operator.