Free Convolution with a Semicircular Distribution and Eigenvalues of Spiked Deformations of Wigner Matrices

By M. Capitaine, Catherine Donati-Martin, Delphine F'eral, Maxime Février. 2010

82 citations recorded by the source corpus.

We investigate the asymptotic spectrum of spiked perturbations of Wigner matrices. The entries of the Wigner matrix have a distribution which is symmetric and satisfies a Poincare inequality. The spectral measure of the deterministic Hermitian perturbation matrix converges to some probability measure with compact support. We also assume that this perturbation matrix has a fixed number of fixed eigenvalues (spikes) outside the support of its limiting spectral measure whereas the distance between the other eigenvalues and this support uniformly goes to zero as the dimension goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of the deformed model, which will converge to some limiting points outside the support of the limiting spectral measure. This phenomenon can be fully described in terms of free probability involving the subordination function related to the free additive convolution by a semicircular distribution. Note that only finite rank perturbations had been considered up to now (even in the deformed GUE case).

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

View Paper: https://www.semanticscholar.org/paper/4485949286b276d1ea27e347f7ae68f72d29d2e8

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

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Citations in the corpus

Outliers in the spectrum of large deformed unitarily invariant models (2012, methodology): “This paper extends the results of [12] to perturbations of arbitrary rank, and it also extends the free probabilistic interpretation of outlier phenomena in terms of subordination functions as described in [20] for deformations of Wigner models.”
A large deviation principle for Wigner matrices without Gaussian tails (2012, background): “nary parts of Yij are independent. We also allow the mean of the entries to be nonzero. Note that the rate 1/ √ nin Theorem A.1 is not necessarily optimal with stronger assumptions; see, for example, [12], equation (3.8). We postpone the proof of Theorem A.1 to the next subsection. We ﬁrst check tha”
The spectral norm of random inner-product kernel matrices (2015, background): “1 of [16] imply ‖ √ γ(ν−a2) p O TWO + a nΛ‖ → ‖μa,ν,γ‖ almost surely.”
Outlier Eigenvalues for Deformed I.I.D. Random Matrices (2014, background): “Note that motivated by the seminal article of Baik, Ben Arous and Péché [6], previous works are devoted to the study of the outlier eigenvalues of deformed random Hermitian matrix models, see notably [4, 7, 8, 9, 10, 19, 21, 22, 23, 28, 34, 35, 36, 37, 41, 43, 45, 46].”
The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, methodology): “Ultimately, we shall apply the Green function comparison method to expressions of the form hv ; G. /vi (step (iv) below).”