Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gaussian Waves and Edge Eigenvectors of Random Regular Graphs

Yukun He, Jiaoyang Huang, Horng-Tzer Yau

TL;DR

The paper addresses the edge behavior of eigenvectors and eigenvalues for random $d$-regular graphs, proving that edge eigenvectors converge to Gaussian waves with unit variance and that edge eigenvalues and eigenvectors are asymptotically independent. It introduces a direct Green’s-function framework based on local resampling, linking the weak convergence of the imaginary part of the resolvent to eigenvector limits and enabling a joint Airy$_1$–Gaussian-wave description. The results extend edge universality to eigenvector coefficients and establish a framework that could apply to other sparse or structured random graphs. This work provides a graph-theoretic analogue of Berry’s random wave conjecture at the spectral edge and sharpens our understanding of spectral data in sparse random environments.

Abstract

Backhausz and Szegedy (2019) demonstrated that the almost eigenvectors of random regular graphs converge to Gaussian waves with variance $0\leq σ^2\leq 1$. In this paper, we present an alternative proof of this result for the edge eigenvectors of random regular graphs, establishing that the variance must be $σ^2=1$. Furthermore, we show that the eigenvalues and eigenvectors are asymptotically independent. Our approach introduces a simple framework linking the weak convergence of the imaginary part of the Green's function to the convergence of eigenvectors, which may be of independent interest.

Gaussian Waves and Edge Eigenvectors of Random Regular Graphs

TL;DR

The paper addresses the edge behavior of eigenvectors and eigenvalues for random -regular graphs, proving that edge eigenvectors converge to Gaussian waves with unit variance and that edge eigenvalues and eigenvectors are asymptotically independent. It introduces a direct Green’s-function framework based on local resampling, linking the weak convergence of the imaginary part of the resolvent to eigenvector limits and enabling a joint Airy–Gaussian-wave description. The results extend edge universality to eigenvector coefficients and establish a framework that could apply to other sparse or structured random graphs. This work provides a graph-theoretic analogue of Berry’s random wave conjecture at the spectral edge and sharpens our understanding of spectral data in sparse random environments.

Abstract

Backhausz and Szegedy (2019) demonstrated that the almost eigenvectors of random regular graphs converge to Gaussian waves with variance . In this paper, we present an alternative proof of this result for the edge eigenvectors of random regular graphs, establishing that the variance must be . Furthermore, we show that the eigenvalues and eigenvectors are asymptotically independent. Our approach introduces a simple framework linking the weak convergence of the imaginary part of the Green's function to the convergence of eigenvectors, which may be of independent interest.

Paper Structure

This paper contains 22 sections, 21 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Topological statements
  4. Proof ideas
  5. Outline of the Paper
  6. Notation
  7. Preliminaries
  8. Kesten-Mckay Distribution
  9. Local Resampling
  10. Green's function extension with general weights
  11. Green's function Estimates
  12. Edge Universality and Optimal Rigidity
  13. Gaussian Waves and Edge Eigenvectors
  14. Green's Function of Switched Graphs
  15. Proof of \ref{['t:ev_universality']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Fix $d\geqslant 3$, and set ${\mathcal{A}}=d(d-1)/(d-2)^2$. Fix $k\geqslant 1$ and radius $r\geqslant 0$. Let ${A}$ be the adjacency matrix of random $d$-regular graphs ${\mathcal{G}}$ on $N$ vertices, and set $H\mathrel{\mathop:}= {A}/\sqrt{d-1}$. Let $\lambda_1 = {d}/{\sqrt{d-1}} \geqslant \lambda where $A_1, A_2, \cdots$ are the points of the Airy$_1$ point process, and $Z_1, Z_2,\cdots, Z_{k}$

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Lemma 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: huang2024spectrum
  • Definition 2.5
  • ...and 32 more