Paper: arXiv 2509.25076

Authors: Volodymyr Riabov, Konstantin Tikhonov, Jean-Philippe Bouchaud

Abstract

We compute exactly the overlap between the eigenvectors of two large empirical covariance matrices computed over intersecting time intervals, generalizing the results obtained previously for non-intersecting intervals. Our method relies on a particular form of Girko linearisation and extended local laws. We check our results numerically and apply them to financial data.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical generalization of eigenvector overlaps in Random Matrix Theory (RMT) for overlapping time periods using Girko linearization and multi-resolvent local laws. While the empirical validation on US stock returns demonstrates practical utility for detecting non-stationarity, the empirical section functions primarily as a toy illustration rather than an exhaustive backtest.

Research Flowchart

  flowchart TD
    A[Research Goal: Compute eigenvector overlaps of sample covariance matrices with intersecting time periods] --> B[Data Inputs: Synthetic Random Matrices & Empirical Financial Data]
    B --> C[Methodology: Girko Linearisation & Extended Local Laws]
    C --> D[Computational Process: Derive exact asymptotic overlap formula & run numerical simulations]
    D --> E[Outcome 1: Exact analytical solution generalizing non-intersecting time intervals]
    D --> F[Outcome 2: Validated cross-asset structure and noise reduction in financial data]

Cited References (21)

  1. The Dyson equation with linear self-energy: spectral bands, edges and cusps (2020)
  2. Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices (2024)
  3. Elementary excitation modes in a granular glass above jamming (2010)
  4. Uncovering functional signature in neural systems via random matrix theory (2019)
  5. Scaling and renormalization in high-dimensional regression (2024) — link
  6. Nonlinear random matrix theory for deep learning (2017)
  7. A random matrix approach to neural networks (2018)
  8. Eigenvector dynamics: general theory and some applications (2012)
and 13 more
  1. Optimal multi-resolvent local laws for Wigner matrices (2022)
  2. Eigenvector overlaps in large sample covariance matrices and nonlinear shrinkage estimators (2024) — link
  3. Eigenstate thermalization hypothesis for Wigner-type matrices (2024) — link
  4. Cusp universality for correlated random matrices (2024) — link
  5. Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices (2024)
  6. Overlaps between eigenvectors of correlated random matrices (2018) — link
  7. Eigenstate thermalization hypothesis for Wigner matrices (2021) — link
  8. Kendall Correlation Coefficients for Portfolio Optimization — → 2410.17366
  9. A first course in random matrix theory: for physicists, engineers and data scientists (2020)
  10. The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropy (2007) — link
  11. Excess Out-of-Sample Risk and Fleeting Modes (2022) — link
  12. Eigenvectors of some large sample covariance matrix ensembles (2011) — link
  13. Distribution of eigenvalues for some sets of random matrices (1967) — link

1 of 21 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.