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Delocalized eigenvectors of transitive graphs and beyond

Nicolas Burq, Cyril Letrouit

TL;DR

The paper develops a spectral-projection framework to establish delocalization of eigenvectors for vertex-transitive graphs and to extend delocalization to approximate eigenvectors of general symmetric matrices. By analyzing the spectral projector $\Pi_I$ and its diagonal, the authors derive sharp $L^q$ bounds for eigenvectors and, under two structural assumptions—few short loops (BST) and Green-function bounds—prove strong delocalization for approximate eigenvectors in windows containing many eigenvalues. The results include Gaussian statistics for entries of random eigenbases in large eigenspaces and optimal $L^\infty$ bounds for approximate eigenvectors of random lifts, with broad applicability to products of graphs and non-homogeneous settings. The work unifies and extends prior results in Cayley graphs, random matrices, and quantum ergodicity on manifolds, providing robust tools for analyzing delocalization across graph models. Overall, delocalization emerges as a universal phenomenon for large, structured graphs under modest spectral and combinatorial conditions.

Abstract

We prove delocalization of eigenvectors of vertex-transitive graphs via elementary estimates of the spectral projector. We recover in this way known results which were formerly proved using representation theory. Similar techniques show that for general symmetric matrices, most approximate eigenvectors spectrally localized in a given window containing sufficiently many eigenvalues are delocalized in $L^q$ norms. Building upon this observation, we prove a delocalization result for approximate eigenvectors of large graphs containing few short loops, under an assumption on the resolvent which is verified in some standard cases, for instance random lifts of a fixed base graph.

Delocalized eigenvectors of transitive graphs and beyond

TL;DR

The paper develops a spectral-projection framework to establish delocalization of eigenvectors for vertex-transitive graphs and to extend delocalization to approximate eigenvectors of general symmetric matrices. By analyzing the spectral projector and its diagonal, the authors derive sharp bounds for eigenvectors and, under two structural assumptions—few short loops (BST) and Green-function bounds—prove strong delocalization for approximate eigenvectors in windows containing many eigenvalues. The results include Gaussian statistics for entries of random eigenbases in large eigenspaces and optimal bounds for approximate eigenvectors of random lifts, with broad applicability to products of graphs and non-homogeneous settings. The work unifies and extends prior results in Cayley graphs, random matrices, and quantum ergodicity on manifolds, providing robust tools for analyzing delocalization across graph models. Overall, delocalization emerges as a universal phenomenon for large, structured graphs under modest spectral and combinatorial conditions.

Abstract

We prove delocalization of eigenvectors of vertex-transitive graphs via elementary estimates of the spectral projector. We recover in this way known results which were formerly proved using representation theory. Similar techniques show that for general symmetric matrices, most approximate eigenvectors spectrally localized in a given window containing sufficiently many eigenvalues are delocalized in norms. Building upon this observation, we prove a delocalization result for approximate eigenvectors of large graphs containing few short loops, under an assumption on the resolvent which is verified in some standard cases, for instance random lifts of a fixed base graph.
Paper Structure (33 sections, 20 theorems, 127 equations)

This paper contains 33 sections, 20 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Delocalized random eigenbases of vertex-transitive graphs
  4. Delocalized approximate eigenvectors of general symmetric matrices
  5. Stronger delocalization of approximate eigenvectors under two assumptions
  6. Assumption of few short loops
  7. Assumption on the Green functions
  8. Statement of the result
  9. Proof techniques and a measure of delocalization
  10. Related results
  11. Erdös-Rényi and regular graphs.
  12. Wigner and Lévy matrices.
  13. Cayley graphs.
  14. "Non-homogeneous" graphs and matrices.
  15. Compact Riemannian manifolds.
  16. ...and 18 more sections

Key Result

Theorem 1.1

There exists $C>0$ such that the following holds. Let $n\in\mathbb{N}^*$ and let $G$ be a vertex-transitive graph with $n$ vertices. Then for any $\Lambda>0$, with probability $\geq 1-n^{2-\log(\Lambda)}$ on the choice of an element $\mathcal{B}\in \mathcal{B}(G)$ picked following $\nu$, any $u\in\m Also, there exists $C>0$ such that for any $q\in[2,+\infty)$ and any $\Lambda>0$, with probability

Theorems & Definitions (42)

  • Theorem 1.1: $L^q$-delocalized eigenbases
  • Theorem 1.2: Gaussian statistics
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 32 more