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Extremal eigenvectors of sparse random matrices

Yukun He, Jiaoyang Huang, Chen Wang

TL;DR

This work establishes edge-eigenvector universality for a broad class of sparse random matrices $A=H+f\,\boldsymbol{e}\boldsymbol{e}^T$ with $H$ real-symmetric and sparse, covering the Erdős–Rényi adjacency model after normalization. Central to the analysis is an isotropic local semicircle law in the sparse regime, proved via a novel cumulant-expansion framework and a systematized abstract polynomial of Green functions that tracks complex dependencies. The paper directly computes the joint distributions of non-trivial edge eigenvectors in deterministic orthogonal directions, showing asymptotic joint Gaussianity and, consequently, edge universality in both eigenvalues and eigenvectors. Applications include normal fluctuations in quantum ergodicity at the spectral edge for Wigner matrices and the extension of edge universality to sparse random graphs. A companion treatment extends the method to complex Wigner matrices, reinforcing the robustness of the approach across architectures of randomness and enabling broader universality conclusions.

Abstract

We consider a class of sparse random matrices, which includes the adjacency matrix of Erdős-Rényi graph ${\bf G}(N,p)$. For $N^{-1+o(1)}\leq p\leq 1/2$, we show that the non-trivial edge eigenvectors are asymptotically jointly normal. The main ingredient of the proof is an algorithm that directly computes the joint eigenvector distributions, without comparisons with GOE. The method is applicable in general. As an illustration, we also use it to prove the normal fluctuation in quantum ergodicity at the edge for Wigner matrices. Another ingredient of the proof is the isotropic local law for sparse matrices, which at the same time improves several existing results.

Extremal eigenvectors of sparse random matrices

TL;DR

This work establishes edge-eigenvector universality for a broad class of sparse random matrices with real-symmetric and sparse, covering the Erdős–Rényi adjacency model after normalization. Central to the analysis is an isotropic local semicircle law in the sparse regime, proved via a novel cumulant-expansion framework and a systematized abstract polynomial of Green functions that tracks complex dependencies. The paper directly computes the joint distributions of non-trivial edge eigenvectors in deterministic orthogonal directions, showing asymptotic joint Gaussianity and, consequently, edge universality in both eigenvalues and eigenvectors. Applications include normal fluctuations in quantum ergodicity at the spectral edge for Wigner matrices and the extension of edge universality to sparse random graphs. A companion treatment extends the method to complex Wigner matrices, reinforcing the robustness of the approach across architectures of randomness and enabling broader universality conclusions.

Abstract

We consider a class of sparse random matrices, which includes the adjacency matrix of Erdős-Rényi graph . For , we show that the non-trivial edge eigenvectors are asymptotically jointly normal. The main ingredient of the proof is an algorithm that directly computes the joint eigenvector distributions, without comparisons with GOE. The method is applicable in general. As an illustration, we also use it to prove the normal fluctuation in quantum ergodicity at the edge for Wigner matrices. Another ingredient of the proof is the isotropic local law for sparse matrices, which at the same time improves several existing results.

Paper Structure

This paper contains 24 sections, 33 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cumulant expansion
  4. Proof of Theorem \ref{['thm1.3']}
  5. Proof of \ref{['3.2']}
  6. Abstract polynomial of Green functions
  7. Proof of Lemma \ref{['lemma 3.5']}
  8. Proofs of \ref{['3.2.2']} and \ref{['3.3']}
  9. Proof of \ref{['3.2.2']}
  10. Proof of \ref{['3.3']}
  11. Applications of Theorem \ref{['thm1.3']}
  12. Proof of Theorem \ref{['thmbulk']}
  13. Proof of Theorem \ref{['thm1.4']}
  14. Proof of Theorem \ref{['thm1.2']}
  15. More properties of the random edge
  16. ...and 9 more sections

Key Result

Theorem 1.2

Fix $k\in \mathbb N_+$, and let $\boldsymbol{\mathrm{v}}_1,\boldsymbol{\mathrm{w}}_1,...,\boldsymbol{\mathrm{v}}_k,\boldsymbol{\mathrm{w}}_k \in \mathbb S^{N-1}_\perp$ be deterministic. Fix $T>0$. There exists fixed $\varepsilon>0$ such that uniformly for all $t_a\in [-T,T]$. Here $(\boldsymbol{\mathrm{z}}_a)_{a=1}^{k}$ are i.i.d. standard Gaussian vectors in $\mathbb R^{N}$.

Theorems & Definitions (50)

  • Definition 1.1: Sparse matrix
  • Theorem 1.2: Universality of edge eigenvectors
  • Remark 1.3
  • Theorem 1.4: Isotropic local law
  • Corollary 1.5
  • Theorem 1.6: Universality of bulk eigenvectors
  • Remark 1.7
  • Theorem 1.8: Edge universality
  • Theorem 1.9
  • Remark 1.10
  • ...and 40 more