Limits of spiked random matrices I
By Alex Bloemendal, B'alint Vir'ag. Probability theory and related fields, 2010
148 citations recorded by the source corpus.
Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general β analogue, proving a conjecture of Baik et al. (Ann Probab 33:1643–1697, 2005). We also treat shifted mean Gaussian orthogonal and β ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schrödinger operator on the half-line, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which β appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at β = 2,4, yielding in particular a new and simple proof of the Painlevé representations for these Tracy–Widom distributions.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/ac9a0e99053d5f56967dfe752acd1022239cef3a
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Citations in the corpus
- The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, methodology): “Subsequently, the results of [3, 31] were extended to the other Gaussian ensembles, such as GOE (the Gaussian orthogonal ensemble), by Bloemendal and Virág [9, 10].”
- Universality of the Stochastic Airy Operator (2013, methodology): “As for convergence of Hn to H , the needed result from [20], as extended in [3], reads:”
- Singular vector and singular subspace distribution for the matrix denoising model (2018, background): “The fluctuation of the extreme eigenvalues of various models have been considered in [2, 3, 7, 8, 9, 23, 24, 28, 14, 15, 42, 55, 34, 56, 59, 44].”
- Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, background): “The equivalent of the above mentioned phase transition were obtained in the real setting in [7, 8, 29, 23].”
- Asymptotic power of sphericity tests for high-dimensional data (2013, background): “At the threshold, the null and the alternative hypotheses lead to different asymptotic distributions for the centered and normalized largest sample covariance eigenvalue [Bloemendal and Virág (2012) and Mo (2012)], which implies some asymptotic detection power.”
- On the principal components of sample covariance matrices (2014, methodology): “The proofs of [3,34] use an asymptotic analysis of Fredholm determinants, while those of [11,12] use an explicit tridiagonal representation of X X∗; both of these approaches rely heavily on the Gaussian nature of X .”
- Universality for the largest eigenvalue of sample covariance matrices with general population (2013, methodology): “On spiked Wishart model, one can refer to [3] for classical complex case and [7, 8, 26, 29, 38] for real case.”