arXiv 4 Aug 2026 · Mathematics — Probability
arXiv:2608.04162 · PDF · Extracted main text
The set of $n\times n$ correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate $\exp{-\tfrac14 n^2\log n}$. 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 $\sqrt{\log n/n}$, its Frobenius distance is asymptotic to $\sqrt n$, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact $\operatorname{Beta}(1,d)$ distribution, where $d=n(n-1)/2$, and is therefore of order $n^{-2}$. More generally, distinct off-diagonal entries are exactly pairwise independent under every $\operatorname{LKJ}(η)$ law. For the uniform law, this yields a Chen-Stein proof of the extreme-correlation point-process limit and an $O(n^{-1})$ total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-$n$ means. We also identify two distinct scales: $η_n\asymp n$ alters the limiting spectrum, whereas $η_n\asymp n^2$ is needed to keep the Frobenius distance bounded. Finally, for a bounded, centered i.i.d. off-diagonal specification, projection to the nearest correlation matrix incurs a squared repair cost asymptotically at least one-half of the squared Frobenius norm of its off-diagonal part.
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| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Daniel Lewandowski, Dorota Kurowicka, and Harry Joe (2009) Generating random correlation matrices based on vines and extended onion method | 0.941 | 6 | 4 | 83% |
| 2 | Ilya Archakov and Peter Reinhard Hansen (2021) A new parametrization of correlation matrices self | 0.928 | 4 | 3 | 100% |
| 3 | Harry Joe (2006) Generating random correlation matrices based on partial correlations | 0.855 | 8 | 6 | 62% |
| 4 | Johannes Heiny, Thomas Mikosch, and Jorge Yslas (2021) Point process convergence for the off-diagonal entries of sample covariance matrices | 0.843 | 3 | 3 | 100% |
| 5 | Anca M. Hanea and Gabriela F. Nane (2018) The asymptotic distribution of the determinant of a random correlation matrix | 0.794 | 6 | 3 | 50% |
| 6 | C. R. Johnson and G. Nvdal (1998) The probability that a (partial) matrix is positive semidefinite | 0.737 | 3 | 3 | 67% |
| 7 | Walter Böhm and Kurt Hornik (2014) Generating random correlation matrices by the simple rejection method: Why it does not work | 0.737 | 3 | 2 | 100% |
| 8 | Tiefeng Jiang (2004) The limiting distributions of eigenvalues of sample correlation matrices | 0.644 | 4 | 2 | 50% |
| 9 | Ilya Archakov, Peter Reinhard Hansen, and Yiyao Luo (2024) A new method for generating random correlation matrices self | 0.644 | 2 | 2 | 100% |
| 10 | John Barnard, Robert McCulloch, and Xiao-Li Meng (2000) Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage | 0.644 | 2 | 2 | 100% |
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