Scaling in the eigenvalue fluctuations of the empirical correlation matrices

TL;DR

This paper investigates the eigenvalue fluctuation scaling in empirical correlation matrices using spacing ratios, providing a new approach that simplifies analysis and applies to real-world data like stock markets and atmospheric measurements.

Contribution

It introduces a novel scaling relation for spacing ratios in eigenvalue spectra, applicable to Wishart matrices and empirical data, avoiding the need for unfolding procedures.

Findings

Spacing ratio distributions follow a predictable scaling relation.

The scaling relation is validated on stock market and atmospheric data.

Analytical and numerical evidence support the proposed scaling law.

Abstract

The spectra of empirical correlation matrices, constructed from multivariate data, are widely used in many areas of sciences, engineering and social sciences as a tool to understand the information contained in typically large datasets. In the last two decades, random matrix theory-based tools such as the nearest neighbour eigenvalue spacing and eigenvector distributions have been employed to extract the significant modes of variability present in such empirical correlations. In this work, we present an alternative analysis in terms of the recently introduced spacing ratios, which does not require the cumbersome unfolding process. It is shown that the higher order spacing ratio distributions for the Wishart ensemble of random matrices, characterized by the Dyson index $\beta$, is related to the first order spacing ratio distribution with a modified value of co-dimension $\beta'$. This scaling is demonstrated for Wishart ensemble and also for the spectra of empirical correlation matrices drawn from the observed stock market and atmospheric pressure data. Using a combination of analytical and numerics, such scalings in spacing distributions are also discussed.