6 results
Search Results
Now showing 1 - 6 of 6
- ItemLocalization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model([Madralin] : EMIS ELibEMS, 2018) Flegel, FranziskaWe 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]<∞}<1/4, 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 γc=1/4 is sharp. Indeed, other recent results imply that for γ>1/4 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.
- ItemEigenvector localization in the heavy-tailed random conductance model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Flegel, FranziskaWe generalize our former localization result about the principal Dirichlet eigenvector of the i.i.d. heavy-tailed random conductance Laplacian to the first k eigenvectors. We overcome the complication that the higher eigenvectors have fluctuating signs by invoking the Bauer-Fike theorem to show that the kth eigenvector is close to the principal eigenvector of an auxiliary spectral problem.
- ItemHomogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Flegel, Franziska; Heida, Martin; Slowik, MartinWe study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for the normalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian’s Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence.
- ItemOn quenched homogenization of long-range random conductance models on stationary ergodic point processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Heida, MartinWe study the homogenization limit on bounded domains for the long-range random conductance model on stationary ergodic point processes on the integer grid. We assume that the conductance between neares neighbors in the point process are always positive and satisfy certain weight conditions. For our proof we use long-range two-scale convergence as well as methods from numerical analysis of finite volume methods.
- ItemA central limit theorem for the effective conductance: I. Linear boundary data and small ellipticity contrasts(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Biskup, Marek; Salvi, Michele; Wolff, TilmanWe consider resistor networks on Zd where each nearest-neighbor edge is assigned a non-negative random conductance. Given a finite set with a prescribed boundary condition, the effective conductance is the minimum of the Dirichlet energy over functions that agree with the boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box is known to converge to a deterministic limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) central limit theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and arbitrary ellipticity contrasts are to be addressed in a subsequent paper.
- ItemLocalization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Flegel, FranziskaWe 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.