Critical tensor covariance at the Marchenko-Pastur threshold

Abstract

For a centered, variance-one random variable $X$ with finite fourth moment, let $x$ be the vector of square-free degree-$d$ monomials in $n$ independent copies of $X$. At the critical scale $d^2/n\to \lambda \in (0,\infty )$, the normalized squared length of $x$ converges to the lognormal variable $R=\exp (\sqrt{\lambda v}Z-\lambda v/2)$, where $v=\mathbb{E}{X}^4-1$ and $Z$ is standard normal. If $\left(\genfrac{}{}{0ex}{}{n}{d}\right)/N\to c\in (0,\infty )$, the sample covariance of $N$ independent copies of $x$ has an almost-sure limiting spectral law: the free compound-Poisson law with rate $1/c$ and jump distribution $\mathrm{Law}(cR)$. It reduces to Marchenko-Pastur when $\lambda v=0$. The proof shows that subtracting the contribution of the sample length leaves vanishing quadratic-form fluctuations, even when $\mathbb{E}{X}^3\ne 0$; length and direction need not be independent.