Spectrum and Lifshitz tails for the Anderson model on the Sierpinski gasket graph

TL;DR

This paper investigates the spectral properties of the Anderson model on the Sierpinski gasket graph, establishing the spectrum, density of states, and Lifshitz tails, revealing how fractal geometry influences localization phenomena.

Contribution

It determines the spectrum, proves the existence of the integrated density of states, and characterizes Lifshitz tails for the Anderson model on the Sierpinski gasket, linking fractal dimensions to spectral behavior.

Findings

Spectrum is almost surely identified.

Existence of the integrated density of states is proven.

Lifshitz tails with exponents related to fractal dimensions are shown.

Abstract

In this work, we study the Anderson model on the Sierpinski gasket graph. We first identify the almost sure spectrum of the Anderson model when the support of the random potential has no gaps. We then prove the existence of the integrated density states of the Anderson model and show that it has Lifshitz tails with Lifshitz exponent determined by the ratio of the volume growth rate and the random walk dimension of the Sierpinski gasket graph.