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09:30 AM - 10:30 AM
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Sparse regular random graphs: spectral density and eigenvectors
Ioana Dumitriu (University of California, San Diego)
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- Adjacency matrices of regular random graphs are a good example of non-Wigner ensembles for which the semicircle law still holds, in various regimes. The one we focus on is when the degree is polylogarithmic in the number of vertices (a "sparse" case). We show that the empirical spectral distribution converges to the semicircle law, estimate the rate of convergence (also known as the "local semicircle law"), and show some results that point toward the delocalization and lack of bias for the second through last eigenvectors.
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10:30 AM - 11:00 AM
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Tea
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11:00 AM - 12:00 PM
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The Single Ring Theorem
Alice Guionnet (École Normale Supérieure de Lyon)
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- I will discuss the asymptotics of the spectrum of non-normal random matrices, whose law is invariant under the action of the orthogonal or the unitary group. This is based on a joint work with M. Krishnapur and O. Zeitouni.
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12:00 PM - 02:00 PM
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Lunch
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02:00 PM - 03:00 PM
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Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices
Mireille Capitaine (Centre National de la Recherche Scientifique (CNRS))
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- We investigate the asymptotic behavior of the eigenvalues of spiked perturbations of Wigner matrices when the dimension goes to infinity. The perturbation matrix is a deterministic Hermitian matrix whose spectral measure converges to some probability measure with compact support. We 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 additive free convolution of the limiting spectral measure of the perturbation matrix by a semi-circular distribution.
This is a joint work with C. Donati-Martin, D. Féral and M. Février.
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03:00 PM - 03:30 PM
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Tea
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03:30 PM - 04:30 PM
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Central limit theorem for linear eigenvalue statistics of diluted random matrices
Mariya Shcherbina (B. Verkin Institute for Low Temperature Physics)
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- We discuss the linear eigenvalue statistics of large random graph in the regimes when the mean number of edges for each vertex tends to infinity. We prove that for the test functions with two derivatives the fluctuations of linear eigenvalue statistics converges in distribution to the Gaussian random variable with zero mean and the variance which coincides with "non gaussian" part of the Wigner ensemble variance.
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