High-dimensional sample covariance matrices with Curie-Weiss entries

Abstract

We study the limiting spectral distribution of sample covariance matrices $X{X}^T$, 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 Marčenko-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.