Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fluctuations in Quantum Unique Ergodicity at the Spectral Edge

Lucas Benigni, Nixia Chen, Patrick Lopatto, Xiaoyu Xie

TL;DR

This work resolves a gap in the understanding of quantum unique ergodicity fluctuations for Wigner matrices at the spectral edge. It proves a central limit theorem for edge eigenvectors and for edge observables of the form $\langle u,Au\rangle$ with traceless $A$ and a lower bound on $\text{Tr}(A^2)$, by a non-dynamical two-moment matching to the Gaussian ensemble. The authors develop a regularization scheme for QUE observables, introduce a novel multi-resolvent local law at the edge, and execute a resolvent-based comparison to GOE, augmented by a polynomialization technique to control higher moments. The results yield Gaussian fluctuations for edge mass distributions on large index sets and for general edge observables, advancing the QUE program for chaotic quantum systems and strengthening the link between Wigner matrices and Gaussian ensembles in extreme spectral regimes.

Abstract

We study the eigenvector mass distribution of an $N\times N$ Wigner matrix on a set of coordinates $I$ satisfying $| I | \ge c N$ for some constant $c >0$. For eigenvectors corresponding to eigenvalues at the spectral edge, we show that the sum of the mass on these coordinates converges to a Gaussian in the $N \rightarrow \infty$ limit, after a suitable rescaling and centering. The proof proceeds by a two moment matching argument. We directly compare edge eigenvector observables of an arbitrary Wigner matrix to those of a Gaussian matrix, which may be computed explicitly.

Fluctuations in Quantum Unique Ergodicity at the Spectral Edge

TL;DR

This work resolves a gap in the understanding of quantum unique ergodicity fluctuations for Wigner matrices at the spectral edge. It proves a central limit theorem for edge eigenvectors and for edge observables of the form with traceless and a lower bound on , by a non-dynamical two-moment matching to the Gaussian ensemble. The authors develop a regularization scheme for QUE observables, introduce a novel multi-resolvent local law at the edge, and execute a resolvent-based comparison to GOE, augmented by a polynomialization technique to control higher moments. The results yield Gaussian fluctuations for edge mass distributions on large index sets and for general edge observables, advancing the QUE program for chaotic quantum systems and strengthening the link between Wigner matrices and Gaussian ensembles in extreme spectral regimes.

Abstract

We study the eigenvector mass distribution of an Wigner matrix on a set of coordinates satisfying for some constant . For eigenvectors corresponding to eigenvalues at the spectral edge, we show that the sum of the mass on these coordinates converges to a Gaussian in the limit, after a suitable rescaling and centering. The proof proceeds by a two moment matching argument. We directly compare edge eigenvector observables of an arbitrary Wigner matrix to those of a Gaussian matrix, which may be computed explicitly.
Paper Structure (21 sections, 31 theorems, 260 equations)

This paper contains 21 sections, 31 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Related Works
  4. Proof Strategy
  5. Outline
  6. Acknowledgments
  7. Preliminaries
  8. Conventions
  9. Notations and Definitions
  10. Local law for resolvent and multi-resolvent
  11. Central Limit Theorem for GOE
  12. Eigenvector Thermalization
  13. Regularized observables
  14. Definitions
  15. Two moment comparison and proof of the main theorem
  16. ...and 6 more sections

Key Result

Theorem 1.3

Let $H$ be a Wigner matrix and fix $\tau\in(0,1)$. Then there exists $\delta=\delta(\tau)\in(0,1)$ such that the following holds. Let $A=A_N\in\mathbb R^{N\times N}$ be a deterministic sequence of traceless matrices such that $A =A^*$, $\|A\|\leqslant1$, and $\operatorname{Tr}(A^2)\geqslant N^{1-\de with convergence in distribution. Here $\mathcal{N}(0,1)$ is a standard real Gaussian random variab

Theorems & Definitions (74)

  • Definition 1.1: Wigner matrix
  • Remark 1.2
  • Theorem 1.3: Central Limit Theorem
  • Definition 2.1: Stochastic domination
  • Theorem 2.2: Isotropic local law
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7: CipErdSch22optimal
  • ...and 64 more