Phase transition for the smallest eigenvalue of high-dimensional sample correlation matrices
Abstract
We study the smallest nonzero eigenvalue of the sample correlation matrix Rn formed from a pn × n data matrix with i.i.d. real entries ξ of mean zero and unit variance, in the high-dimensional regime pn / n → ϕ∈ (0, ∞) ∖ {1}. In the tall regime ϕ> 1, we prove almost-sure convergence of λn (Rn) to the lower Marčenko--Pastur edge λ- = (1 − √ϕ)2 without additional moment assumptions. In the wide regime ϕ< 1, we establish a phase transition at the third-order tail scale. If t3 ℙ{lvert ξrvert > t} → 0 as t → ∞, the smallest eigenvalue λpn (Rn) converges in probability to λ-. While if t3 ℙ{lvert ξrvert > t} → ∞, then λpn (Rn) converges in probability to zero. At the critical scale t3 ℙ{lvert ξrvert > t} → κ∈ (0, ∞), the point process of eigenvalues in the lower gap (0, λ-) converges in distribution to a Poisson random measure with explicit intensity. In this critical regime, we also identify the nondegenerate limiting distribution of λpn (Rn), which has a continuous density on (0, λ-) and a positive atom at λ-.
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BibTeX
@misc{lin2026phase,
title = {{Phase transition for the smallest eigenvalue of high-dimensional sample correlation matrices}},
author = {Zeqin Lin and Guamgming Pan and Haozhu Zhao and Wang Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15731},
eprint = {2609.15731},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15731},
}
RIS
TY - UNPB
TI - Phase transition for the smallest eigenvalue of high-dimensional sample correlation matrices
AU - Zeqin Lin
AU - Guamgming Pan
AU - Haozhu Zhao
AU - Wang Zhou
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15731
UR - https://arxiv.org/abs/2609.15731
AB - We study the smallest nonzero eigenvalue of the sample correlation matrix R_n formed from a p_n × n data matrix with i.i.d. real entries ξ of mean zero and unit variance, in the high-dimensional regime p_n / n → ϕ∈ (0, ∞) ∖ {1}. In the tall regime ϕ> 1, we prove almost-sure convergence of λ_n (R_n) to the lower Marčenko--Pastur edge λ_- = (1 − √ϕ)^2 without additional moment assumptions. In the wide regime ϕ< 1, we establish a phase transition at the third-order tail scale. If t^3 ℙ{lvert ξrvert > t} → 0 as t → ∞, the smallest eigenvalue λ_p_n (R_n) converges in probability to λ_-. While if t^3 ℙ{lvert ξrvert > t} → ∞, then λ_p_n (R_n) converges in probability to zero. At the critical scale t^3 ℙ{lvert ξrvert > t} → κ∈ (0, ∞), the point process of eigenvalues in the lower gap (0, λ_-) converges in distribution to a Poisson random measure with explicit intensity. In this critical regime, we also identify the nondegenerate limiting distribution of λ_p_n (R_n), which has a continuous density on (0, λ_-) and a positive atom at λ_-.
ER -