Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Laura Shou, Wei Wang, Shiwen Zhang

Abstract

In this work, we study the Anderson model on graphs with Ahlfors $α$-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 $α$-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.

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

Abstract

In this work, we study the Anderson model on graphs with Ahlfors -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 to Ahlfors -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.

Paper Structure

This paper contains 9 sections, 15 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lifshitz Tails Imply Fractional Moment Bounds for the Green's Function
  3. Lifshitz Tail and Initial Estimate
  4. Geometric Decoupling
  5. Depleted Green Function Estimate and the Inductive Step
  6. Lifshitz Tail Estimate
  7. (Modified) Neumann Bracketing and the Upper Bound
  8. (Modified) Dirichlet bracketing and the lower bound
  9. Combes--Thomas estimates, A priori bounds, and Decoupling inequalities

Key Result

Theorem 1.1

Let $H_\omega$ be the Anderson model given in eqn:AM on a graph $(\mathbb{V}, \mathcal{E})$ satisfying eqn:vol-control for some $\alpha > 0$. Assume that: Then: $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: The unit triangle $T_0$ is located next to the origin, with vertices $\mathbb{V}_0=\{(0,0),(1,0),(1/2,\sqrt3/2)\}$. The right hand side of the $y$-axis is the 3rd step of construction $T_3$ and the left hand side is its reflective mirror with respect to the $y$-axis. The picture contains $2\times 27$ many unit triangles $T$, which are all translations of $T_0$. The dots form the vertex set $\mathbb{V}_3'\cup\mathbb{V}_3$. The edge set $\mathcal{E}_3'\cup\mathcal{E}_3$ consists of all edges of length 1 of the unit triangles.
  • Figure 2: The first two generations of the Sierpinski tetrahedron (Sierpinski pyramid) graph. The infinite graph has volume dimension $\alpha_3=\log4/\log2$ and walk dimension $\beta_3=\log 6/\log2$.

Theorems & Definitions (31)

  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.5
  • ...and 21 more