High-dimensional sample covariance matrices with Curie–Weiss entries

Abstract

We study the limiting spectral distribution of sample covariance matrices XX T , where X are p × n random matrices with correlated entries and p/n → y ∈ [0, ∞). 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 β > 0. The model exhibits a phase transition at β = 1. The correlation between any two entries is of order O((np) −1 ) for β ∈ (0, 1), O((np) −1/2 ) for β = 1, and for β > 1 the correlation does not vanish in the limit. In our proofs we use Stieltjes transforms and concentration of random quadratic forms.

Thin-shell theory for rotationally invariant random simplices

“…To the best of our knowledge, the most general setting in which the limiting distribution of the log-volume (or equivalently the log-determinant) was derived was in [9,60] who assumed that the entries of Y possess a finite fourth moment, which is the typical assumption in papers on linear spectral statistics. We refer to [8,18,27,28,21,11,29,30] for collections of results which show the stark differences in the asymptotic behavior under infinite fourth moments.…”

Thin-shell theory for rotationally invariant random simplices

“…To the best of our knowledge, the most general setting in which the limiting distribution of the log-volume (or equivalently the log-determinant) was derived was in [9,59] who assumed that the entries of Y possess a finite fourth moment, which is the typical assumption in papers on linear spectral statistics. We refer to [8,18,27,28,21,11,29,30] for collections of results which show the stark differences in the asymptotic behavior under infinite fourth moments.…”

Large Sample Covariance Matrices of Gaussian Observations with Uniform Correlation Decay

“…In other words, some covariance structures -such as the equicovariant structure (in Theorem 3 below) -cannot be expressed as a Kronecker product. Another study where correlations may span both across rows and columns is [8], where the authors derived the Marchenko-Pastur law for data matrices filled with jointly correlated random spins stemming from the Curie-Weiss model from statistical physics. There, even for the case of non-vanishing correlations, the Marchenko-Pastur law is recovered.…”