Anderson-like Transition for a Class of Random Sparse Models in d >= 2 Dimensions

TL;DR

This paper demonstrates a spectral transition in a class of high-dimensional random sparse models, showing a shift from absolutely continuous to pure point spectrum, akin to Anderson localization phenomena.

Contribution

It introduces a new high-dimensional sparse model exhibiting Anderson-like spectral transition, extending understanding of localization in complex systems.

Findings

Spectral transition from absolutely continuous to pure point spectrum

Applicable to models in dimensions d >= 2

Potential implications for physics and open problems

Abstract

We show that the Kronecker sum of d >= 2 copies of a random one-dimensional sparse model displays a spectral transition of the type predicted by Anderson, from absolutely continuous around the center of the band to pure point around the boundaries. Possible applications to physics and open problems are discussed briefly.