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Eigenvector decorrelation for random matrices

Giorgio Cipolloni, László Erdős, Joscha Henheik, Oleksii Kolupaiev

TL;DR

The paper analyzes how eigenvectors of two random, deformed Wigner matrices decorrelate under perturbations. It develops a two-resolvent local law for products of resolvents $G_1 A G_2$ with a deterministic approximation $M_{12}^A$, and introduces a stability framework with a time-dependent control parameter and a linear LT term to capture deformation- and energy-difference effects. It then proves two main results: (i) an ETH-type decomposition for regular observables and an overlap-based ETH for general observables, and (ii) an optimal bound on eigenvector overlaps that decays with the deformation distance $igra (D_1-D_2)^2ig a$ and the energy differences, leading to strong decorrelation when the deformations diverge or spectra separate. The analysis hinges on a zigzag strategy combining a global law, a characteristic-flow zig, and a Green function comparison zag, complemented by a rigorous stability analysis and careful handling of regular observables. The results extend ETH-type eigenvector statistics to pairs of spectral families and quantify decorrelation in a broad random-matrix setting with deterministic deformations, offering new insights into eigenvector behavior under perturbations relevant for quantum chaos and high-dimensional statistics.

Abstract

We study the sensitivity of the eigenvectors of random matrices, showing that even small perturbations make the eigenvectors almost orthogonal. More precisely, we consider two deformed Wigner matrices $W+D_1$, $W+D_2$ and show that their bulk eigenvectors become asymptotically orthogonal as soon as $\mathrm{Tr}(D_1-D_2)^2\gg 1$, or their respective energies are separated on a scale much bigger than the local eigenvalue spacing. Furthermore, we show that quadratic forms of eigenvectors of $W+D_1$, $W+D_2$ with any deterministic matrix $A\in\mathbf{C}^{N\times N}$ in a specific subspace of codimension one are of size $N^{-1/2}$. This proves a generalization of the Eigenstate Thermalization Hypothesis to eigenvectors belonging to two different spectral families.

Eigenvector decorrelation for random matrices

TL;DR

The paper analyzes how eigenvectors of two random, deformed Wigner matrices decorrelate under perturbations. It develops a two-resolvent local law for products of resolvents with a deterministic approximation , and introduces a stability framework with a time-dependent control parameter and a linear LT term to capture deformation- and energy-difference effects. It then proves two main results: (i) an ETH-type decomposition for regular observables and an overlap-based ETH for general observables, and (ii) an optimal bound on eigenvector overlaps that decays with the deformation distance and the energy differences, leading to strong decorrelation when the deformations diverge or spectra separate. The analysis hinges on a zigzag strategy combining a global law, a characteristic-flow zig, and a Green function comparison zag, complemented by a rigorous stability analysis and careful handling of regular observables. The results extend ETH-type eigenvector statistics to pairs of spectral families and quantify decorrelation in a broad random-matrix setting with deterministic deformations, offering new insights into eigenvector behavior under perturbations relevant for quantum chaos and high-dimensional statistics.

Abstract

We study the sensitivity of the eigenvectors of random matrices, showing that even small perturbations make the eigenvectors almost orthogonal. More precisely, we consider two deformed Wigner matrices , and show that their bulk eigenvectors become asymptotically orthogonal as soon as , or their respective energies are separated on a scale much bigger than the local eigenvalue spacing. Furthermore, we show that quadratic forms of eigenvectors of , with any deterministic matrix in a specific subspace of codimension one are of size . This proves a generalization of the Eigenstate Thermalization Hypothesis to eigenvectors belonging to two different spectral families.

Paper Structure

This paper contains 37 sections, 23 theorems, 280 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Previous related results
  4. Multi--resolvent local laws
  5. The method of characteristics
  6. Main results
  7. First main result: Regular observables and eigenstate thermalization (Theorem \ref{['theo:maintheo2']})
  8. Second main result: Optimal eigenvector decorrelation (Theorem \ref{['theo:maintheo']})
  9. Proofs of the Main Results: Multi-resolvent local laws
  10. Preliminaries on the stability operator
  11. Multi--resolvent local law: Proofs of Theorems \ref{['theo:maintheo']} and \ref{['theo:maintheo2']}
  12. Optimality and possible extensions of Theorem \ref{['theo:multigllaw']}
  13. Zigzag strategy: Proof of Theorem \ref{['theo:multigllaw']}
  14. Input global laws
  15. Zig step: Propagating bounds via the characteristic flow
  16. ...and 22 more sections

Key Result

Theorem 2.4

Fix any $\kappa>0$ and fix $D_1,D_2\in\mathbf{C}^{N\times N}$ with $\lVert D_l\rVert\lesssim 1$. Let $W$ be a Wigner matrix satisfying Assumption ass:momass, and, for $l=1,2$, denote by ${\bm u}_1^l,\dots, {\bm u}_N^l$ the orthonormal eigenvectors of $W+D_l$. Fix indices $i,j$ such that the quantile More generally, for arbitrary observables $A\in\mathbf{C}^{N\times N}$, we have where $V=V(\nu_1,\

Figures (1)

  • Figure 1: In gray, we illustrated the $\Im z > 0$ part of the bulk-restricted spectral domains $\Omega_{\kappa,t}$ for three times, $t= 0$, $t \in (0,T)$, and $t = T$ (the $\Im z < 0$ part is obtained by reflection). On each of the panels, the graph of the density $\rho_t$ is superimposed in a dash-dotted style. The solid curve in the $t=T$ panel represents the implicitly defined curve $\vert\Im z\vert \rho(z)=N^{-1+\epsilon}$, above which one has the unrestricted domain $\Omega_T \supset \Omega_{\kappa, T}$. On the same $t = T$ panel, the region below the dashed curve is removed in the rhs. of \ref{['eq:specdom_bulk']}. For $t<T$ the solid and the dashed curves are the images of the corresponding curves at $t=T$ under the flow $\mathfrak{F}_{t,T}$.

Theorems & Definitions (43)

  • Definition 2.2: Regular observables
  • Remark 2.3: On the choice of $V$
  • Theorem 2.4: Generalized Eigenstate Thermalization
  • Example 2.5: Eigenstate Thermalization
  • Theorem 2.6: Optimal eigenvector decorrelation
  • Remark 2.7: Eigenvector correlation in perturbative regime
  • Remark 2.8: Independence of eigenvalue gaps
  • Proposition 3.1: Stability bound
  • Theorem 3.2: Average two-resolvent local laws in the bulk
  • Lemma 3.3: Comparison of different regularizations
  • ...and 33 more