Large deviations of the extreme eigenvalues of random deformations of matrices

By Florent Benaych-Georges, Alice Guionnet, M. Maida. Probability theory and related fields, 2010

59 citations recorded by the source corpus.

Consider a real diagonal deterministic matrix Xn of size n with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale n, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of Xn converge to the edges of the support of the limiting measure and when we allow some eigenvalues of Xn, that we call outliers, to converge out of the bulk. We can also generalise our results to the case when Xn is random, with law proportional to e−n Tr V(X) dX, for V growing fast enough at infinity and any perturbation of finite rank.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9d783c26ff670318070d75d3c17f8737ef732c4b

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In Discrete Mathematics and Combinatorics

1. Jun Yin
2. Antti Knowles
3. Horng-Tzer Yau
4. L. Erdős
5. Terence Tao
6. Ji Oon Lee
7. Van H. Vu
8. Debashis Paul

In Mathematical Physics

1. Jun Yin
2. Horng-Tzer Yau
3. Antti Knowles
4. L. Erdős
5. Ji Oon Lee
6. Terence Tao
7. Xiucai Ding
8. Debashis Paul

In Statistics and Probability

1. Jun Yin
2. Horng-Tzer Yau
3. Antti Knowles
4. L. Erdős
5. Xiucai Ding
6. Ji Oon Lee
7. Terence Tao
8. Alexander Aue

Citations in the corpus

• Outliers in the spectrum of iid matrices with bounded rank perturbations (2010, background): “ j=1 ( j( p1 n X n) z) whenever the denominator is nonzero. We are indebted to Alice Guionnet for suggesting the use of this type of criterion, versions of which also appear in [3], [8], [18], [13], [14], [15]. For instance, the k= 1 case of this criterion appears explicitly in [13]. In [3] the expr”
• On the principal components of sample covariance matrices (2014, background): “Previously, eigenvalue sticking was established for a certain class of random perturbations of Wigner matrices in [6,7].”
• The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, methodology): “We use the following representation of the eigenvalues of z H , which was already used in several papers on finite-rank deformations of random matrices [5, 6, 7, 32].”
• Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails (2015, background): “Then in [6], the large deviations for the joint law of the extreme eigenvalues of a deterministic real diagonal matrix perturbed with a low rank Hermitian matrix with delocalized eigenvectors are studied extensively.”
• Random matrix theory in statistics: A review (2014, background): “Benaych-Georges and Nadakuditi (2011) gave a more intuitive explanation of the phase transition phenomenon that brings out the central role played by quadratic forms involving the resolvent of a null Wishart matrix.”
• Statistical thresholds for tensor PCA (2018, background): “ttention in the random matrix theory community. Far richer information is also known, such as universality, large deviations, and uctuation theorems. For a small sample of work in this direction, see [42, 11, 14, 13]. More recently, it has been shown that the BBP transition is also the transition fo”