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Localization of One-Dimensional Random Band Matrices

Reuben Drogin

TL;DR

The paper proves exponential localization of eigenvectors for a broad class of one-dimensional random band matrices in the regime $W^2\ll n$, with decay on the scale $W^2$. By combining this localization result with recent delocalization findings, it establishes a sharp localization-delocalization transition governed by the scaling $W^2/n$. The proof hinges on a Schenker-type resolvent decomposition, a conditional fluctuation analysis with a Mermin–Wagner perturbation to generate $W^{-1/2}$-scale fluctuations, and a Wegner estimate to control spectral probabilities, thereby bridging detailed probabilistic structure with resolvent methods. These results advance the rigorous understanding of RBMs and their spectral statistics, connecting to Anderson localization phenomena and the broader RBM/delocalization landscape.

Abstract

We consider a general class of $n\times n$ random band matrices with bandwidth $W$. When $W^2\ll n$, we prove that with high probability the eigenvectors of such matrices are localized and decay exponentially at the sharp scale $W^2$. Combined with the delocalization results of Yau and Yin [arXiv:2501.01718] and of Erdős and Riabov [arXiv:2506.06441], this establishes the conjectured localization-delocalization transition for a large class of random band matrices.

Localization of One-Dimensional Random Band Matrices

TL;DR

The paper proves exponential localization of eigenvectors for a broad class of one-dimensional random band matrices in the regime , with decay on the scale . By combining this localization result with recent delocalization findings, it establishes a sharp localization-delocalization transition governed by the scaling . The proof hinges on a Schenker-type resolvent decomposition, a conditional fluctuation analysis with a Mermin–Wagner perturbation to generate -scale fluctuations, and a Wegner estimate to control spectral probabilities, thereby bridging detailed probabilistic structure with resolvent methods. These results advance the rigorous understanding of RBMs and their spectral statistics, connecting to Anderson localization phenomena and the broader RBM/delocalization landscape.

Abstract

We consider a general class of random band matrices with bandwidth . When , we prove that with high probability the eigenvectors of such matrices are localized and decay exponentially at the sharp scale . Combined with the delocalization results of Yau and Yin [arXiv:2501.01718] and of Erdős and Riabov [arXiv:2506.06441], this establishes the conjectured localization-delocalization transition for a large class of random band matrices.

Paper Structure

This paper contains 14 sections, 11 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Model and Results
  3. Background and Motivation
  4. Prior Mathematical Work
  5. Proof Sketch
  6. Further Directions and Open Questions
  7. Notation and Conventions
  8. Acknowledgements
  9. Schenker Decomposition and Wegner Estimate
  10. Fluctuations Lower Bound
  11. Proof of Theorem \ref{['thm:single-energy-decay']}
  12. Appendix
  13. Proof of Lemma \ref{['eq:elementary-jensens-assum']}
  14. Proof of Wegner Estimate

Key Result

Theorem 1

Let $M,E_0>0$ and $H$ be as in eq:definition-of-H with independent entries up to the constraint $H=H^{\ast}$. If the nonzero entries of $\sqrt{W}H$ are $M$-regular, and $\left(\psi_j, E_j\right)_{j\in \left[1,NW\right]}$ are the normalized eigenvector-eigenvalue pairs of $H$, then for any $N,W>0$ an The constants $C,c>0$ depend only on $M$ and $E_0$.

Theorems & Definitions (24)

  • Definition 1: M-Regular
  • Theorem 1
  • Theorem 2
  • Conjecture 1
  • Proposition 1
  • proof
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • proof
  • ...and 14 more