High-dimensional sample covariance matrices with Curie-Weiss entries

TL;DR

This paper analyzes the spectral distribution of high-dimensional sample covariance matrices with correlated entries modeled as Curie-Weiss spins, revealing phase transitions and different limiting behaviors depending on the correlation strength.

Contribution

It introduces a study of spectral distributions for covariance matrices with Curie-Weiss correlated entries, identifying phase transitions and asymptotic behaviors based on correlation regimes.

Findings

For $y>0$, the spectral distribution converges to Marčenko-Pastur.

For $y=0$, the distribution converges to the semicircle law after rescaling.

Correlation decay rates depend on the inverse temperature $eta$, with phase transition at $eta=1$.

Abstract

We study the limiting spectral distribution of sample covariance matrices $XX^T$, where $X$ are $p\times n$ random matrices with correlated entries, for the cases $p/n\to y\in [0,\infty)$. If $y>0$, we obtain the Mar\v{c}enko-Pastur distribution and in the case $y=0$ the semicircle distribution (after appropriate rescaling). The entries we consider are Curie-Weiss spins, which are correlated random signs, where the degree of the correlation is governed by an inverse temperature $\beta>0$. The model exhibits a phase transition at $\beta=1$. The correlation between any two entries decays at a rate of $O(np)$ for $\beta \in (0,1)$, $O(\sqrt{np}$) for $\beta=1$, and for $\beta>1$ the correlation does not vanish in the limit. In our proofs we use Stieltjes transforms and concentration of random quadratic forms.