Correlation Matrices in High Dimensions: Volume, Spectrum, and Extremes
Abstract
The set of n× n correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate exp{−1/4 n2log n}. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular. Its maximum absolute correlation is of order √(log n/n), its Frobenius distance is asymptotic to √n, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact Beta(1,d) distribution, where d=n(n−1)/2, and is therefore of order n−2. Distinct off-diagonal entries are exactly pairwise independent under every LKJ(η) law, which yields a Chen-Stein proof of the extreme-correlation point-process limit for the whole LKJ family and an O(n−1) total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-n means. Finally, for a bounded, centered i.i.d. off-diagonal perturbation of any correlation matrix, the nearest-correlation projection removes a fraction of the squared perturbation that tends to one and recovers the original matrix in average squared error per entry.
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BibTeX
@misc{hansen2026correlation,
title = {{Correlation Matrices in High Dimensions: Volume, Spectrum, and Extremes}},
author = {Peter Reinhard Hansen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.04162},
eprint = {2608.04162},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.04162},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Correlation Matrices in High Dimensions: Volume, Spectrum, and Extremes
AU - Peter Reinhard Hansen
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.04162
UR - https://arxiv.org/abs/2608.04162
AB - The set of n× n correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate exp{−1/4 n^2log n}. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular. Its maximum absolute correlation is of order √(log n/n), its Frobenius distance is asymptotic to √n, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact Beta(1,d) distribution, where d=n(n−1)/2, and is therefore of order n^−2. Distinct off-diagonal entries are exactly pairwise independent under every LKJ(η) law, which yields a Chen-Stein proof of the extreme-correlation point-process limit for the whole LKJ family and an O(n^−1) total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-n means. Finally, for a bounded, centered i.i.d. off-diagonal perturbation of any correlation matrix, the nearest-correlation projection removes a fraction of the squared perturbation that tends to one and recovers the original matrix in average squared error per entry.
ER -