Spectral and Dynamical contrast on highly correlated Anderson-type models

TL;DR

This paper investigates the spectral and dynamical properties of correlated Anderson-type models on specific two-dimensional graphs, revealing how geometric modifications and high disorder induce contrasting behaviors and phase transitions within their spectra.

Contribution

It introduces and analyzes Anderson models with long-range correlated potentials on two-dimensional graphs, highlighting the impact of geometry and disorder on spectral and dynamical phenomena.

Findings

Geometric changes and high disorder significantly affect spectral properties.

The vertical model shows a sharp phase transition within its absolutely continuous spectrum.

Contrasting behaviors are observed between diagonal and vertical models.

Abstract

We study spectral and dynamical properties of random Schr\"odinger operators $H_{\mathrm{Vert}}=-A_{\mathbb{G}_{\mathrm{Vert}}}+V_{\omega}$ and $H_{\mathrm{Diag}}=-A_{\mathbb{G}_{\mathrm{Diag}}}+V_{\omega}$ on certain two dimensional graphs ${\mathbb{G}_{\mathrm{Vert}}}$ and ${\mathbb{G}_{\mathrm{Diag}}}$. Differently from the standard Anderson model, the random potentials are not independent but, instead, are constant along any vertical line, i.e $V_{\omega}(n)=\omega(n_1)$, for $n=(n_1,n_2)$. In particular, the potentials studied here exhibit long range correlations. We present examples where geometric changes to the underlying graph, combined with high disorder, have a significant impact on the spectral and dynamical properties of the operators, leading to contrasting behaviors for the "diagonal" and "vertical" models. Moreover, the "vertical" model exhibits a sharp phase transition within its (purely) absolutely continuous spectrum. This is captured by the notions of transient and recurrent components of the absolutely continuous spectrum, introduced by Avron and Simon.