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

Yunfeng Shi, Li Wen, Dongfeng Yan

TL;DR

This work establishes spectral localization for a random operator on $\mathbb{Z}^d$ with long-range hopping decaying as $e^{-\\log^{\rho}(\|\bm x\|+1)}$ ($\rho>1$) and Hölder-continuous disorder. By adapting multi-scale analysis to the nonlocal hopping via a quasi-metric Green's-function framework and a Wegner estimate for Hölder distributions, the authors prove pure point spectrum and eigenfunctions that decay at the same rate as the hopping, for sufficiently small $\varepsilon$ (large disorder). The results provide a partial resolution of Yeung and Oono's conjecture in higher dimensions and extend the MSA toolkit to long-range random operators. The combination of Green's-function bounds, scale induction, and Shnol-type arguments yields robust localization on a full energy interval for a.e. disorder.

Abstract

In this paper, we investigate random operators on $\mathbb{Z}^d$ with Hölder continuously distributed potentials and the long-range hopping. The hopping amplitude decays with the inter-particle distance $\|\bm x\|$ as $e^{-\log^ρ(\|\bm x\|+1)}$ with $ρ>1,\bm x\in\Z^d$. 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].

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

TL;DR

This work establishes spectral localization for a random operator on with long-range hopping decaying as () and Hölder-continuous disorder. By adapting multi-scale analysis to the nonlocal hopping via a quasi-metric Green's-function framework and a Wegner estimate for Hölder distributions, the authors prove pure point spectrum and eigenfunctions that decay at the same rate as the hopping, for sufficiently small (large disorder). The results provide a partial resolution of Yeung and Oono's conjecture in higher dimensions and extend the MSA toolkit to long-range random operators. The combination of Green's-function bounds, scale induction, and Shnol-type arguments yields robust localization on a full energy interval for a.e. disorder.

Abstract

In this paper, we investigate random operators on with Hölder continuously distributed potentials and the long-range hopping. The hopping amplitude decays with the inter-particle distance as with . 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].

Paper Structure

This paper contains 12 sections, 14 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. The Notations
  3. Main results
  4. The initial step
  5. The Wegner Estimate
  6. Iteration Theorem
  7. The validity of $({\bf P2})_s$
  8. Coupling Lemma and the Proof of Theorems \ref{['thm1']}
  9. Coupling Lemma
  10. Proof of Theorem \ref{['ite']}
  11. Proof of Theorem \ref{['thm1']}
  12. Proof of theorem \ref{['thm2']}

Key Result

Theorem 3.4

Let $\mu\in\mathscr{H}(\lambda)$ (i.e., $\mathcal{D}_{\lambda}(\mu)>0$), $E_0\in\mathbb R$, $L_0\in \mathbb{N}$ and $L_{s+1}=[L_s^{\alpha}]$ ($s\geq 0$). Then for $0<\beta<\mathcal{D}_{\lambda}(\mu)$, there exists such that the following holds true. For $L_0\ge\underline{L}_0$, there are $\varepsilon_0=\varepsilon_0(\lambda,\mu,\beta,d,\gamma,\rho, L_0)>0$ and $\eta=\eta(\lambda,\mu,\beta,d,L_0)>

Theorems & Definitions (37)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3: cf. CKM87
  • Theorem 3.4
  • Theorem 3.5
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Lemma 5.1: Wegner estimate, cf. Theorem 6.2 of CKM87
  • ...and 27 more