Abstract
It is well known that an \(n \times n\) Wishart matrix with d degrees of freedom is close to the appropriately centered and scaled Gaussian orthogonal ensemble (GOE) if d is large enough. Recent work of Bubeck, Ding, Eldan, and Racz, and independently Jiang and Li, shows that the transition happens when \(d = \Theta ( n^{3} )\). Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when \(d / n^{3} \rightarrow c \in (0, \infty )\). This shows, in particular, that the phase transition from Wishart to GOE is smooth.
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References
Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010)
Anderson, G.W., Zeitouni, O.: A CLT for a band matrix model. Probab. Theory Relat. Fields 134(2), 283–338 (2006)
Bubeck, S., Ding, J., Eldan, R., Rácz, M.Z.: Testing for high-dimensional geometry in random graphs. Random Struct. Algorithms 49(3), 503–532 (2016)
Bubeck, S., Ganguly S.: Entropic CLT and phase transition in high-dimensional Wishart matrices (2015). Preprint. http://arxiv.org/abs/1509.03258
Devroye, L., György, A., Lugosi, G., Udina, F.: High-dimensional random geometric graphs and their clique number. Electron. J. Probab. 16, 2481–2508 (2011)
El Karoui, N.: On the largest eigenvalue of Wishart matrices with identity covariance when \(n\), \(p\) and \(p/n \rightarrow \infty \) (2003). arXiv:math/0309355
El Karoui, N.: Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. Ann. Probab. 35, 663–714 (2007)
Eldan, R., Mikulincer, D.: Information and dimensionality of anisotropic random geometric graphs (2016). Preprint. https://arxiv.org/abs/1609.02490
Jiang, T., Li, D.: Approximation of rectangular beta-Laguerre ensembles and large deviations. J. Theor. Probab. 28, 804–847 (2015)
Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)
Johnstone, I.M.: On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat. 29(2), 295–327 (2001)
Wishart, J.: The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A(1/2), 32–52 (1928)
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We thank an anonymous reviewer for helpful suggestions.
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Rácz, M.Z., Richey, J. A Smooth Transition from Wishart to GOE. J Theor Probab 32, 898–906 (2019). https://doi.org/10.1007/s10959-018-0808-2
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DOI: https://doi.org/10.1007/s10959-018-0808-2
Keywords
- Random matrix theory
- Wishart distribution
- Gaussian Orthogonal Ensemble (GOE)
- Total variation
- Phase transition