Localization for random operators on $\mathbb{Z}^d$ with the long-range hopping

TL;DR

This paper proves that random operators on multi-dimensional integer lattices with long-range hopping and high disorder exhibit pure point spectrum and localized eigenfunctions, extending understanding of localization phenomena in such systems.

Contribution

It demonstrates localization for random operators with long-range hopping using multi-scale analysis, addressing a conjecture from Yeung and Oono.

Findings

Pure point spectrum established at large disorder

Eigenfunctions decay at the same rate as the hopping term

Partial validation of a long-standing conjecture

Abstract

In this paper, we investigate random operators on $\mathbb{Z}^d$ with H\"older continuously distributed potentials and the long-range hopping. The hopping amplitude decays with the inter-particle distance $\|\bm x\|$ as $e^{-\log^{\rho}(\|\bm x\|+1)}$ with $\rho>1,\bm x\in\Z^d$. By employing the multi-scale analysis (MSA) technique, we prove that for large disorder, the random operators have pure point spectrum with localized eigenfunctions whose decay rate is the same as the hopping term. This gives a partial answer to a conjecture of Yeung and Oono [{\it Europhys. Lett.} 4(9), (1987): 1061-1065].