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Spectrum of Random $d$-regular Graphs Up to the Edge

Jiaoyang Huang, Horng-Tzer Yau

TL;DR

This paper establishes sharp spectral statistics for random $d$-regular graphs with fixed degree. By developing a Green’s function framework built around a self-consistent boundary weight $Q({\mathcal G},z)$ and a carefully controlled tree-extension model, the authors prove eigenvalue rigidity to the Kesten–McKay locations and polynomial-edge concentration $\lambda_2, | lambda_N| \le 2+N^{-c}$, together with complete eigenvector delocalization for all eigenvectors. The core methodology relies on a novel local resampling (switching) mechanism that preserves uniformity and enables exchangeable pair arguments, enabling fluctuation averaging and robust Green’s-function control down to scales near the spectral edge. This yields a high-probability local law and edge-corrected rigidity that generalize prior sparse-matrix results to the edge regime, with optimal-like error terms. The work provides a flexible toolbox for sparse random graph spectra, including stability under local changes and concentration phenomena, with implications for Ramanujan-type behavior and quantum ergodicity on sparse graphs.

Abstract

Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq3$. We prove that, with probability $1-N^{-1+{\varepsilon}}$ for any ${\varepsilon} >0$, the following two properties hold as $N \to \infty$ provided that $d\geq3$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten-McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in $N$, i.e. $λ_2, |λ_N|\leq 2+N^{-c}$. (ii) All eigenvectors of random $d$-regular graphs are completely delocalized.

Spectrum of Random $d$-regular Graphs Up to the Edge

TL;DR

This paper establishes sharp spectral statistics for random -regular graphs with fixed degree. By developing a Green’s function framework built around a self-consistent boundary weight and a carefully controlled tree-extension model, the authors prove eigenvalue rigidity to the Kesten–McKay locations and polynomial-edge concentration , together with complete eigenvector delocalization for all eigenvectors. The core methodology relies on a novel local resampling (switching) mechanism that preserves uniformity and enables exchangeable pair arguments, enabling fluctuation averaging and robust Green’s-function control down to scales near the spectral edge. This yields a high-probability local law and edge-corrected rigidity that generalize prior sparse-matrix results to the edge regime, with optimal-like error terms. The work provides a flexible toolbox for sparse random graph spectra, including stability under local changes and concentration phenomena, with implications for Ramanujan-type behavior and quantum ergodicity on sparse graphs.

Abstract

Consider the normalized adjacency matrices of random -regular graphs on vertices with fixed degree . We prove that, with probability for any , the following two properties hold as provided that : (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten-McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in , i.e. . (ii) All eigenvectors of random -regular graphs are completely delocalized.

Paper Structure

This paper contains 57 sections, 41 theorems, 368 equations, 4 figures.

Table of Contents

  1. Introduction and Main Results
  2. Introduction
  3. Statements of main results: eigenvalue statistics
  4. Delocalization of Eigenvectors
  5. Comments on key ideas
  6. Some notations
  7. Tree-like Graphs
  8. Graphs
  9. Graphs, adjacency matrices, Green's functions
  10. Subsets and Subgraphs
  11. Neighborhoods
  12. Trees
  13. Excess of random $d$-regular graphs
  14. Trees and tree extension
  15. Green's function for extension of graphs with weight D(z)
  16. ...and 42 more sections

Key Result

Theorem 1.2

Fix $d\geqslant 3$, $0<{\mathfrak c}<1$, ${\mathfrak R}=({\mathfrak c}/4)\log_{d-1}N$ and recall the set of radius-${\mathfrak R}$ tree like graphs $\bar{\Omega}\subset {\sf G}_{N,d}$ from Definition def:barOmega. For any large ${\mathfrak C}>0$ and $N$ large enough, with probability $1-\mathop{\mat where $\gamma_i$ are classical eigenvalue locations given by the Kesten-McKay distribution kmlaw

Figures (4)

  • Figure 1: Theorem \ref{['thm:delocalizationev']} shows that a random $d$-regular graph has only completely delocalized eigenvectors with probability $1-\mathop{\mathrm{O}}\nolimits(N^{-(1-\mathop{\mathrm{o}}\nolimits(1))\omega_d})$. On the other hand, it is not difficult to show that a random $d$-regular graph has localized eigenvectors with probability $\Omega(N^{-d+2})$. For example, a random $3$-regular graph contains the subgraph shown above with probability $\Omega(N^{-1})$. Our results imply that the eigenvectors of random $3$-regular graphs are completely delocalized with probability $1-N^{-1+\mathop{\mathrm{o}}\nolimits(1)}$, which is optimal.
  • Figure 2: The sum of weighted paths from $j$ to itself staying outside ${\mathcal{T}}$ is approximately given by $G_{jj}^{(i)}$. In average, $G_{jj}^{(i)}$ can be approximated by the Green's function at root vertex of a depth $\ell$ truncated $(d-1)$-ary tree with boundary weights $Q({\mathcal{G}},z)$.
  • Figure 3: The left figure illustrates a finite graph ${\mathcal{G}}$; its extensible vertices are shown as grey circles. The right figure shows the tree extension $\mathrm{TE}({\mathcal{G}})$, in which a rooted tree (darkly shaded) is attached to every extensible vertex.
  • Figure 4: The switching encoded by the two directed edges $\vec{S}=\{(v_1, v_2), (v_3, v_4)\}$ replaces the unoriented edges $\{v_1,v_2\}, \{v_3,v_4\}$ by $\{v_1,v_4\},\{v_2,v_3\}$.

Theorems & Definitions (112)

  • Definition 1.1
  • Theorem 1.2: Eigenvalue Rigidity
  • Theorem 1.3: Extremal eigenvalues
  • Theorem 1.4: Eigenvector delocalization
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3: deficit function
  • Definition 2.4: tree extenstion
  • ...and 102 more