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 d2/n → λ∈ (0,∞), the normalized squared length of x converges to the lognormal variable R = exp(√(λv) Z − λv/2), where v = 𝔼 X4 − 1 and Z is standard normal. If binomnd/N → c ∈ (0,∞), 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 Law(cR). It reduces to Marchenko-Pastur when λv = 0. The proof shows that subtracting the contribution of the sample length leaves vanishing quadratic-form fluctuations, even when 𝔼 X3 ≠ 0; length and direction need not be independent.
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BibTeX
@misc{xie2026critical,
title = {{Critical tensor covariance at the Marchenko-Pastur threshold}},
author = {Xiaohui Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18293},
eprint = {2608.18293},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18293},
}
RIS
TY - UNPB
TI - Critical tensor covariance at the Marchenko-Pastur threshold
AU - Xiaohui Xie
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.18293
UR - https://arxiv.org/abs/2608.18293
AB - 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 → λ∈ (0,∞), the normalized squared length of x converges to the lognormal variable R = exp(√(λv) Z − λv/2), where v = 𝔼 X^4 − 1 and Z is standard normal. If binomnd/N → c ∈ (0,∞), 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 Law(cR). It reduces to Marchenko-Pastur when λv = 0. The proof shows that subtracting the contribution of the sample length leaves vanishing quadratic-form fluctuations, even when 𝔼 X^3 ≠ 0; length and direction need not be independent.
ER -