Landscape estimates of the integrated density of states for Jacobi operators on graphs

TL;DR

This paper establishes bounds on the integrated density of states for Jacobi operators on various graphs using the localization landscape, providing new insights and proofs for spectral properties like Lifshitz tails.

Contribution

It introduces landscape-based estimates for the integrated density of states on graphs, extending the landscape law to fractal and aperiodic structures, and offers new proofs for spectral phenomena.

Findings

Landscape estimates apply to diverse graph models.

Upper bounds hold for fractal Sierpinski gasket graph.

Landscape law aids in proving Lifshitz tails.

Abstract

We show the integrated density of states for a variety of Jacobi operators on graphs, such as the Anderson model and random hopping models on graphs with Gaussian heat kernel bounds, can be estimated from above and below in terms of the localization landscape counting function. Specific examples of these graphs include stacked and decorated lattices, graphs corresponding to band matrices, and aperiodic tiling graphs. The upper bound part of the landscape law also applies to the fractal Sierpinski gasket graph. As a consequence of the landscape law, we obtain landscape-based proofs of the Lifshitz tails in several models including random band matrix models, certain bond percolation Hamiltonians on $\mathbb{Z}^d$, and Jacobi operators on certain stacks of graphs. We also present intriguing numerical simulations exploring the behavior of the landscape counting function across various models.