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Delocalization of One-Dimensional Random Band Matrices

Horng-Tzer Yau, Jun Yin

TL;DR

This work proves delocalization, quantum diffusion, QUE, and universality for one-dimensional random band matrices with band width above the critical scale W ~ N^{1/2}. The authors develop a stochastic flow and a novel loop hierarchy, introducing G-loops and primitive loops, and provide an explicit tree-based representation of the primitive K-loops. By combining Ward identities, sum-zero properties, and careful scale analysis, they obtain sharp local semicircle laws, eigenvector delocalization bounds, and universality results that align with GUE statistics in the bulk. The loop-analytic framework is built to be adaptable to broader band models and higher dimensions, offering a potentially powerful tool for future studies of localization-delocalization transitions.

Abstract

Consider an $ N \times N$ Hermitian one-dimensional random band matrix with band width $W > N^{1 / 2 + \frak c} $ for any $ {\frak c} > 0$. In the bulk of the spectrum and in the large $ N $ limit, we obtain the following results: (i) The semicircle law holds up to the scale $ N^{-1 + \varepsilon} $ for any $ \varepsilon > 0 $. (ii) All $ L^2 $- normalized eigenvectors are delocalized, meaning their $ L^\infty$ norms are simultaneously bounded by $ N^{-\frac{1}{2} + \varepsilon} $ with overwhelming probability, for any $ \varepsilon > 0 $. (iii) Quantum unique ergodicity holds in the sense that the local $ L^2 $ mass of eigenvectors becomes equidistributed with high probability. (iv) Universality of eigenvalue statistics holds, i.e., the local eigenvalue statistics of these band matrices are given by those of Gaussian unitary ensembles.

Delocalization of One-Dimensional Random Band Matrices

TL;DR

This work proves delocalization, quantum diffusion, QUE, and universality for one-dimensional random band matrices with band width above the critical scale W ~ N^{1/2}. The authors develop a stochastic flow and a novel loop hierarchy, introducing G-loops and primitive loops, and provide an explicit tree-based representation of the primitive K-loops. By combining Ward identities, sum-zero properties, and careful scale analysis, they obtain sharp local semicircle laws, eigenvector delocalization bounds, and universality results that align with GUE statistics in the bulk. The loop-analytic framework is built to be adaptable to broader band models and higher dimensions, offering a potentially powerful tool for future studies of localization-delocalization transitions.

Abstract

Consider an Hermitian one-dimensional random band matrix with band width for any . In the bulk of the spectrum and in the large limit, we obtain the following results: (i) The semicircle law holds up to the scale for any . (ii) All - normalized eigenvectors are delocalized, meaning their norms are simultaneously bounded by with overwhelming probability, for any . (iii) Quantum unique ergodicity holds in the sense that the local mass of eigenvectors becomes equidistributed with high probability. (iv) Universality of eigenvalue statistics holds, i.e., the local eigenvalue statistics of these band matrices are given by those of Gaussian unitary ensembles.
Paper Structure (35 sections, 38 theorems, 657 equations, 14 figures)

This paper contains 35 sections, 38 theorems, 657 equations, 14 figures.

Table of Contents

  1. Introduction
  2. The model and main results
  3. Band matrix model
  4. Main results
  5. Universality
  6. Stochastic flow and G-loops
  7. Propagator Theta and examples of K
  8. Estimates on loops
  9. Strategy of the proofs of main lemmas
  10. Sum zero properties
  11. Notations
  12. Definition and properties of K
  13. Tree representation of K
  14. Ward's identity on K
  15. Sum zero property of K
  16. ...and 20 more sections

Key Result

Theorem 2.2

Suppose that for some $\mathfrak c>0$, For band matrix defined in this subsection we have the following estimate. For any (small) constants $\kappa, \tau>0$ and (large) $D>0$, there exists $N_0$ such that for all $N \geqslant N_0$ we have

Figures (14)

  • Figure 1: Illustration of operator ${\cal G}^{(a)}_{k}$
  • Figure 2: Illustration of operator ${\cal G}^{(a), L}_{k, l}$
  • Figure 3: Illustration of operator ${\cal G}^{(a), R}_{k, l}$
  • Figure 4: Illustration of canonical partition of the polygons
  • Figure 5: Star graph
  • ...and 9 more figures

Theorems & Definitions (90)

  • Definition 2.1: Stochastic domination and high probability event
  • Theorem 2.2: Delocalizaiton
  • Theorem 2.3: Local semicircle law
  • Theorem 2.4: Quantum diffusion
  • proof : Proof of Theorem \ref{['MR:decol']}
  • Theorem 2.5: Generalized quantum unique ergodicity
  • proof : Proof of Theorem \ref{['MR:QUE']}
  • Theorem 2.6: Bulk universality
  • proof : Proof of Theorem \ref{['Thm: B_Univ']}
  • Definition 2.7: The $z_t$ flow
  • ...and 80 more