The distribution of localization centers in some discrete random systems

TL;DR

This paper investigates the distribution of localization centers in discrete random systems, demonstrating convergence to a Poisson process for the Anderson model and infinite divisibility in other models, enhancing understanding of spectral properties.

Contribution

It extends previous work by analyzing the point process of eigenvalues and localization centers, establishing convergence to Poisson and infinite divisibility results in various models.

Findings

Poisson process convergence for the Anderson model

Infinite divisibility of limiting point processes in other models

Enhanced understanding of spectral localization phenomena

Abstract

As a supplement of our previous work, we consider the localized region of the random Schroedinger operators on $l^2({\bf Z}^d)$ and study the point process composed of their eigenvalues and corresponding localization centers. For the Anderson model, we show that, this point process in the natural scaling limit converges in distribution to the Poisson process on the product space of energy and space. In other models with suitable Wegner-type bounds, we can at least show that any limiting point processes are infinitely divisible.