Spectral geometry of flat tori with random impurities

TL;DR

This paper investigates the spectral geometry of flat tori with random impurities, revealing the transition from localized to delocalized eigenfunctions at high energies and providing bounds on localization length.

Contribution

It introduces new results on eigenfunction behavior in disordered flat tori, including delocalization at high energies and bounds on localization length with random impurities.

Findings

Eigenfunctions are localized at the bottom of the spectrum.

High-energy eigenfunctions can be delocalized with positive probability.

A polynomial lower bound for localization length in terms of eigenvalue.

Abstract

We discuss new results on the geometry of eigenfunctions in disor- dered systems. More precisely, we study tori $R^d/LZ^d$, $d=2,3$, with uniformly distributed Dirac masses. Whereas at the bottom of the spectrum eigenfunctions are known to be localized, we show that for sufficiently large eigenvalue there exist uniformly distributed eigenfunctions with positive probability. We also study the limit $L\to\infty$ with a positive density of random Dirac masses, and deduce a lower polynomial bound for the localization length in terms of the eigenvalue for Poisson distributed Dirac masses on $R^d$ . Finally, we discuss some results on the breakdown of localization in random displacement models above a certain energy threshold.