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The Zigzag Strategy for Random Band Matrices

László Erdős, Volodymyr Riabov

TL;DR

The paper proves delocalization for a broad class of one-dimensional random band matrices with bandwidth $W$ satisfying $W \gg \sqrt{N}$ in the bulk spectrum, establishing that eigenvectors are fully delocalized, eigenvalues follow Wigner-Dyson statistics, and QUE holds for general diagonal observables with optimal rates. The authors develop and implement the zigzag strategy, coupling a dynamic characteristic flow with Green function comparisons to derive optimal multi-resolvent local laws (averaged and isotropic) for resolvent chains $G_{[1,k]}$ with diagonal observables, including traceless variants. A key innovation is observable regularization, which mollifies long resolvent chains to control spatial structure and overcome the gapless instability of the self-consistent equations in RBMs. The framework handles general variance profiles and non-Gaussian entries, enabling QUE and Wigner-Dyson universality results beyond special Gaussian block models, and extends insights toward edge behavior and higher-dimensional variants. Overall, the work advances the rigorous understanding of the Anderson delocalized phase for RBMs by delivering sharp local laws, eigenvector statistics, and universality results in a highly general setting, with potential implications for related disordered systems and quantum diffusion phenomena.

Abstract

We prove that a very general class of $N\times N$ Hermitian random band matrices is in the delocalized phase when the band width $W$ exceeds the critical threshold, $W\gg \sqrt{N}$. In this regime, we show that, in the bulk spectrum, the eigenfunctions are fully delocalized, the eigenvalues follow the universal Wigner-Dyson statistics, and quantum unique ergodicity holds for general diagonal observables with an optimal convergence rate. Our results are valid for general variance profiles, arbitrary single entry distributions, in both real-symmetric and complex-Hermitian symmetry classes. In particular, our work substantially generalizes the recent breakthrough result of Yau and Yin [arXiv:2501.01718], obtained for a specific complex Hermitian Gaussian block band matrix. The main technical input is the optimal multi-resolvent local laws -- both in the averaged and fully isotropic form. We also generalize the $\sqrtη$-rule from [arXiv:2012.13215] to exploit the additional effect of traceless observables. Our analysis is based on the zigzag strategy, complemented with a new global-scale estimate derived using the static version of the master inequalities, while the zig-step and the a priori estimates on the deterministic approximations are proven dynamically.

The Zigzag Strategy for Random Band Matrices

TL;DR

The paper proves delocalization for a broad class of one-dimensional random band matrices with bandwidth satisfying in the bulk spectrum, establishing that eigenvectors are fully delocalized, eigenvalues follow Wigner-Dyson statistics, and QUE holds for general diagonal observables with optimal rates. The authors develop and implement the zigzag strategy, coupling a dynamic characteristic flow with Green function comparisons to derive optimal multi-resolvent local laws (averaged and isotropic) for resolvent chains with diagonal observables, including traceless variants. A key innovation is observable regularization, which mollifies long resolvent chains to control spatial structure and overcome the gapless instability of the self-consistent equations in RBMs. The framework handles general variance profiles and non-Gaussian entries, enabling QUE and Wigner-Dyson universality results beyond special Gaussian block models, and extends insights toward edge behavior and higher-dimensional variants. Overall, the work advances the rigorous understanding of the Anderson delocalized phase for RBMs by delivering sharp local laws, eigenvector statistics, and universality results in a highly general setting, with potential implications for related disordered systems and quantum diffusion phenomena.

Abstract

We prove that a very general class of Hermitian random band matrices is in the delocalized phase when the band width exceeds the critical threshold, . In this regime, we show that, in the bulk spectrum, the eigenfunctions are fully delocalized, the eigenvalues follow the universal Wigner-Dyson statistics, and quantum unique ergodicity holds for general diagonal observables with an optimal convergence rate. Our results are valid for general variance profiles, arbitrary single entry distributions, in both real-symmetric and complex-Hermitian symmetry classes. In particular, our work substantially generalizes the recent breakthrough result of Yau and Yin [arXiv:2501.01718], obtained for a specific complex Hermitian Gaussian block band matrix. The main technical input is the optimal multi-resolvent local laws -- both in the averaged and fully isotropic form. We also generalize the -rule from [arXiv:2012.13215] to exploit the additional effect of traceless observables. Our analysis is based on the zigzag strategy, complemented with a new global-scale estimate derived using the static version of the master inequalities, while the zig-step and the a priori estimates on the deterministic approximations are proven dynamically.

Paper Structure

This paper contains 72 sections, 45 theorems, 894 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General background
  3. Informal description of the main results
  4. Informal description of the main methods
  5. History and related results on random band matrices
  6. Notations and conventions
  7. Definition of the model, spatial control functions
  8. Main results: Multi-resolvent local laws and corollaries
  9. Main ideas of the proofs
  10. Development of local laws: Mean-field case
  11. Hierarchy of static master inequalities
  12. Dynamical master inequalities and zigzag
  13. Main ideas for band matrices
  14. Failure of the mean-field approach
  15. Observable regularization
  16. ...and 57 more sections

Key Result

Proposition 2.8

Figures (1)

  • Figure 1: Schematic representation of the Zigzag induction. The random matrices $H^q, V^q$, as defined in \ref{['eq:Vq']}, positioned within an abstract coordinate system. The horizontal axis corresponds to the size of the Gaussian component, while the vertical axis represents the lower bound on $|\mathop{\mathrm{Im}}\nolimits z_t| = \eta_t$ for which the local laws are established. Solid arrows indicate applications of Proposition \ref{['prop:zig']} (referred to as Zig steps), while dashed arrows denote applications of Proposition \ref{['prop:zag']} (Zag steps).

Theorems & Definitions (122)

  • Definition 2.1: Random band matrix (RBM) with bandwidth $W$
  • Example 2.2: Translation-invariant RBM
  • Example 2.3: Block Band Matrix
  • Definition 2.4: Admissible Control Functions
  • Definition 2.5: Admissible variance profile
  • Example 2.6: Profile with Fast Polynomial Decay
  • Example 2.7: Profile with Exponential Decay
  • Proposition 2.8: Admissible $S$
  • Definition 2.9: Model parameters
  • Remark 2.10: Choice of Small Exponents and Large Parameters
  • ...and 112 more