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Random band matrix localization by scalar fluctuations

Nixia Chen, Charles K Smart

TL;DR

The paper addresses the eigenvector localization problem for Gaussian random band matrices, aiming to sharpen the band-width threshold for localization. By reformulating the model as a random block tridiagonal matrix and proving exponential off-diagonal decay of the resolvent via a lower bound on the logarithmic fluctuations of the corner block, the authors show localization when $W \le N^{1/4-ε}$, extending the previous $1/8$-exponent barrier. A central technical achievement is a detailed analysis of scalar fluctuations: the corner resolvent entry is represented as a product of random blocks, and a lower bound on the variance of the logarithm is obtained through a careful study of the conditional law of the block norms $S_k$, leveraging a Markov-cocycle perspective. The results rely on Gaussian structure to control conditional laws, and they provide a near-optimal barrier for this scalar-fluctuation approach, with implications for the Lyapunov-exponent structure of the associated cocycle. Overall, the work advances rigorous understanding of localization in random band matrices and tightens the connection between resolvent decay, fluctuation bounds, and eigenvector localization.

Abstract

We show the eigenvectors of a Gaussian random band matrix are localized when the band width is less than the 1/4 power of the matrix size. Our argument is essentially an optimized version of Schenker's proof of the 1/8 exponent.

Random band matrix localization by scalar fluctuations

TL;DR

The paper addresses the eigenvector localization problem for Gaussian random band matrices, aiming to sharpen the band-width threshold for localization. By reformulating the model as a random block tridiagonal matrix and proving exponential off-diagonal decay of the resolvent via a lower bound on the logarithmic fluctuations of the corner block, the authors show localization when , extending the previous -exponent barrier. A central technical achievement is a detailed analysis of scalar fluctuations: the corner resolvent entry is represented as a product of random blocks, and a lower bound on the variance of the logarithm is obtained through a careful study of the conditional law of the block norms , leveraging a Markov-cocycle perspective. The results rely on Gaussian structure to control conditional laws, and they provide a near-optimal barrier for this scalar-fluctuation approach, with implications for the Lyapunov-exponent structure of the associated cocycle. Overall, the work advances rigorous understanding of localization in random band matrices and tightens the connection between resolvent decay, fluctuation bounds, and eigenvector localization.

Abstract

We show the eigenvectors of a Gaussian random band matrix are localized when the band width is less than the 1/4 power of the matrix size. Our argument is essentially an optimized version of Schenker's proof of the 1/8 exponent.
Paper Structure (8 sections, 9 theorems, 52 equations)

This paper contains 8 sections, 9 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Some context and prior work
  3. Main result
  4. Scalar fluctuations
  5. Discussion of optimality
  6. Acknowledgements
  7. Random full matrix bounds
  8. Fluctuation lower bound

Key Result

Theorem 1.2

If $\varepsilon > 0$, $|\lambda| < \varepsilon^{-1}$, and $N, M \geq C_\varepsilon$, then holds for all $1 \leq i, j \leq N$.

Theorems & Definitions (16)

  • Theorem 1.2
  • Lemma 1.4
  • Conjecture 1.10
  • Lemma 2.1
  • proof
  • Lemma 2.7
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • ...and 6 more