Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model
- Flegel, Franziska
2010 Mathematics Subject Classification
- 47B80 47A75 60J27
- random conductance model, Dirichlet spectrum, eigenfunction localization, heavy tails
We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of Zd (d ≥ 2) with zero Dirichlet condition. We assume that the conductances w are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If γ = sup q ≥ 0; E [w^-q]<∞ <¼, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold γrm c = ¼ is sharp. Indeed, other recent results imply that for γ>¼ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, and path arguments.
- Electron. J. Probab., 23 (2018), pp. 1--43, DOI doi:10.1214/18-EJP160 .