Paper: arXiv 2409.17086

Authors: Elie Attal, Romain Allez

Abstract

We consider the eigenvectors of the principal minor of dimension $n< N$ of the Dyson Brownian motion in $\mathbb{R}^{N}$ and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We explicitly compute the limiting rescaled mean squared overlaps in the large $n,, N$ limit with $n,/,N$ tending to a fixed ratio $q,$, for any initial symmetric matrix $A,$. This is accomplished using a Burgers-type evolution equation for a specific resolvent. In the GOE case, our formula simplifies, and we identify an eigenvector analogue of the well-known interlacing of eigenvalues. We investigate in particular the case where $A$ has isolated eigenvalues. Our method is based on analysing the eigenvector flow under the Dyson Brownian motion.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly sophisticated mathematical framework extending Cauchy’s Interlace Theorem to eigenvectors of large random matrices using Dyson Brownian motion and Burgers-type equations. However, as a purely theoretical probability/RMT paper, it lacks empirical backtesting or real-world quantitative finance data validation, despite brief conceptual mentions of PCA applications.

Research Flowchart

  flowchart TD
    A[Goal: Analyze Eigenvector Overlaps<br>Between Principal Minors & Full Matrix] --> B[Input: Dyson Brownian Motion<br>Initial Symmetric Matrix A]
    B --> C[Derive Eigenvector Flow Equations<br>Under DBM]
    C --> D[Analyze Evolution of Resolvent<br>Via Burgers-type PDE]
    D --> E[Compute Limit as n, N → ∞<br>With Ratio n/N → q]
    E --> F[Outcomes: Explicit Formula for<br>Limiting Mean Squared Overlaps]
    F --> G[Key Findings: Eigenvector Interlacing<br>& Isolated Eigenvalue Dynamics]

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0 of 36 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.