Critical tensor covariance at the Marchenko--Pastur threshold

Abstract

Let $X$ be a centered, variance-one random variable with finite fourth moment, and form the principal degree-$d$ tensor feature vector of all square-free monomials in $n$ independent copies of $X$. For $m$ independent samples we determine the global spectral law of the sample covariance throughout the critical scale $d^2/n\to \lambda \in [0,\infty )$, with aspect ratio $p/m\to c$. For a fixed base distribution with finite fourth moment and $P(|X|=1)<1$, prior work gives ordinary Marchenko--Pastur convergence if and only if $d=o(\sqrt{n})$. We identify the finite critical boundary: when $d^2/n\to \lambda \in (0,\infty )$, the tensor radius converges in quadratic Wasserstein distance to a lognormal law determined by the fourth moment, while all remaining bounded quadratic fluctuations vanish. A leave-one-out resolvent argument then yields almost-sure convergence of the empirical spectral distribution to a free compound-Poisson law driven by this endogenous lognormal jump. The limit reduces to Marchenko--Pastur when the fourth-moment excess or the overlap intensity vanishes. In the unit-modulus case, our estimates recover the sharp range $\min (d,n-d)=o(n)$ for uniform quadratic-form concentration and imply Marchenko--Pastur convergence throughout that range, with an explicit variance bound.