Staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices
Abstract
We establish staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices formed from a pn × n data matrix with i.i.d. real entries of mean zero and unit variance, allowing an infinite fourth moment. In the proportional regime pn / n → ϕ∈ (0, ∞), the first-order asymptotics depend jointly on the aspect ratio and the entry tail. The transitions are driven by collisions of large entries in distinct rows of a common column. The first collision order capable of producing a separated upper outlier is k* (ϕ) = ⌊ √ϕ ⌋ + 2, yielding the critical tail exponent α* (ϕ) = 2 + 2 / k* (ϕ). This exponent decreases in steps as ϕ increases, creating a staircase boundary between convergence to the upper Marčenko--Pastur edge and successive outlier levels. At exact critical tail scales, the point process of eigenvalues above the upper edge or the preceding deterministic level converges to a Poisson point process. The resulting nondegenerate limiting laws for the largest eigenvalue connect adjacent phases and have a positive atom at this baseline. If every fixed collision order is supercritical, the largest eigenvalue diverges in probability despite finite entry variance.
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BibTeX
@misc{li2026staircase,
title = {{Staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices}},
author = {Yanpeng Li and Zeqin Lin and Yiming Liu and Jiahui Xie and Haozhu Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06731},
eprint = {2610.06731},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06731},
}
RIS
TY - UNPB
TI - Staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices
AU - Yanpeng Li
AU - Zeqin Lin
AU - Yiming Liu
AU - Jiahui Xie
AU - Haozhu Zhao
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06731
UR - https://arxiv.org/abs/2610.06731
AB - We establish staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices formed from a p_n × n data matrix with i.i.d. real entries of mean zero and unit variance, allowing an infinite fourth moment. In the proportional regime p_n / n → ϕ∈ (0, ∞), the first-order asymptotics depend jointly on the aspect ratio and the entry tail. The transitions are driven by collisions of large entries in distinct rows of a common column. The first collision order capable of producing a separated upper outlier is k_* (ϕ) = ⌊ √ϕ ⌋ + 2, yielding the critical tail exponent α_* (ϕ) = 2 + 2 / k_* (ϕ). This exponent decreases in steps as ϕ increases, creating a staircase boundary between convergence to the upper Marčenko--Pastur edge and successive outlier levels. At exact critical tail scales, the point process of eigenvalues above the upper edge or the preceding deterministic level converges to a Poisson point process. The resulting nondegenerate limiting laws for the largest eigenvalue connect adjacent phases and have a positive atom at this baseline. If every fixed collision order is supercritical, the largest eigenvalue diverges in probability despite finite entry variance.
ER -