Edge localization and Lifshitz tails for graphs with Ahlfors regular volume growth

TL;DR

This paper extends Lifshitz tail estimates and localization results for the Anderson model from lattices to Ahlfors regular graphs, including fractals like the Sierpinski gasket, under mild regularity assumptions.

Contribution

It generalizes Lifshitz tail and localization results to Ahlfors regular graphs, including fractals, and establishes conditions for spectral and dynamical localization.

Findings

Lifshitz tail estimates are established for Ahlfors regular graphs.

Exponential decay of fractional moments of Green's function implies localization.

Application to Sierpinski gasket shows pure point spectrum and strong dynamical localization.

Abstract

In this work, we study the Anderson model on graphs with Ahlfors $\alpha$-regular volume growth. We show that, under mild regularity assumptions of the random distribution, Lifshitz-tail type estimates near the bottom of the spectrum lead to exponential decay of fractional moments of the Green's function and thus spectral and dynamical localization at low energies. This generalizes the result of [4] from the lattice $\mathbb{Z}^d$ to Ahlfors $\alpha$-regular graphs. In addition, we establish Lifshitz tail estimates for the integrated density of states, with the Lifshitz exponent determined by the ratio of the volume growth rate and the random walk dimension of the underlying graphs, under certain assumptions on low lying eigenvalues of the Dirichlet and Neumann Laplacian on the graph. As an application, we verify all conditions on the Sierpinski gasket graph and obtain that, under mild regularity assumptions of the random distribution, for any fixed disorder, the Anderson model on the Sierpinski gasket graph has pure point spectrum and exhibits strong dynamical localization near the bottom of the spectrum.