High-dimensional sample covariance matrices with Curie-Weiss entries

We study the limiting spectral distribution of sample covariance matrices XXT, where X are p x n random matrices with correlated entries and p/n -> y is an element of [0, infinity). If y > 0, we obtain the Marcenko-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 is of order O((np)(-1)) for beta is an element of (0, 1), O ((np)(-1/2)) 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.