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Yin, Jun

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Yin

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Jun

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Yin, Jun
Author's Publications: search.filters.isAuthorOfPublication.f7815434-3238-40e2-ba1d-b6103e951889

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Now showing 1 - 8 of 8
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    On the principal components of sample covariance matrices
    (Springer Nature, 2015) Bloemendal, Alex; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun
    We introduce a class of M×MM×M sample covariance matrices Q which subsumes and generalizes several previous models. The associated population covariance matrix Σ=[E] Σ=EQ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of Σ−IMΣ−IM may depend on MM in an arbitrary fashion. We investigate the principal components, i.e. the top eigenvalues and eigenvectors, of Q . We derive precise large deviation estimates on the generalized components [] of the outlier and non-outlier eigenvectors [] . Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the subcritical spikes of ΣΣ . The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix Q , satisfying [E] =IMEQ=IM , as input: the isotropic local Marchenko–Pastur law established in Bloemendal et al. (Electron J Probab 19:1–53, 2014), level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue–eigenvector distribution.
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    Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
    (Springer Nature, 2012) Erdos, Laszlo; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun
    We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN≫N2/3pN≫N2/3 , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.
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    Local circular law for random matrices
    (Springer Nature, 2013) Bourgade, Paul; Yau, Horng-Tzer; Yin, Jun
    The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point zz away from the unit circle. More precisely, if ||z|−1|≥τ||z|−1|≥τ for arbitrarily small τ>0τ>0 , the circular law is valid around zz up to scale N−1/2+εN−1/2+ε for any ε>0ε>0 under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.
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    Isotropic local laws for sample covariance and generalized Wigner matrices
    (Institute of Mathematical Statistics, 2014) Alex, Bloemendal; Erdos, Laszlo; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun
    We consider sample covariance matrices of the form X ∗X, where X is an M × N matrix with independent random entries. We prove the isotropic local MarchenkoPastur law, i.e. we prove that the resolvent (X ∗X − z) −1 converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity hv,(X ∗X − z) −1wi − hv, wim(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v, w ∈ C N . We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Im z > N −1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.
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    The local circular law II: the edge case
    (Springer Nature, 2013) Bourgade, Paul; Yau, Horng-Tzer; Yin, Jun
    In the first part of this article (Bourgade et al. arXiv:1206.1449, 2012), we proved a local version of the circular law up to the finest scale N−1/2+εN−1/2+ε for non-Hermitian random matrices at any point z∈ℂz∈C with ||z|−1|>c||z|−1|>c for any c>0c>0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case |z|−1=o(1)|z|−1=o(1). Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge |z|−1=o(1)|z|−1=o(1) up to scale N−1/4+εN−1/4+ε.
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    Fixed Energy Universality for Generalized Wigner Matrices
    (Wiley-Blackwell, 2015) Bourgade, Paul; Erdos, Laszlo; Yau, Horng-Tzer; Yin, Jun
    We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.
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    A Lower Bound on the Ground State Energy of Dilute Bose Gas
    (American Institute of Physics, 2010) Lee, Ji; Yin, Jun
    Consider an N-Boson system interacting via a two-body repulsive short-range potential \({V}\) in a three dimensional box \({\Lambda}\) of side length \({L}\). We take the limit \({N}\), \({L}\) \({\rightarrow}\) \({\infty}\) while keeping the density \({\rho}\) = \({N}\)/\({L}\)\({^3}\) fixed and small. We prove a new lower bound for its ground state energy per particle $$\frac{E(N, \Lambda)}{N} \geq 4 \pi a \rho [ 1 - O(\rho^{1/3} |\log \rho|^3) ],$$ as \({\rho}\)\({\rightarrow}\) 0, where \({a}\) is the scattering length of \({V}\).
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    Bulk universality for generalized Wigner matrices
    (Springer Science + Business Media, 2011) Erdos, Laszlo; Yau, Horng-Tzer; Yin, Jun
    Consider \(N × N\) Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure \(\nu_{ij}\) with a subexponential decay. Let \(\sigma_{ij}^2\) be the variance for the probability measure \(\nu_{ij}\) with the normalization property that \(\sum_i\sigma_{ij}^2 = 1\) for all j. Under essentially the only condition that \(c\leq N\sigma_{ij}^2 \leq c^{−1}\) for some constant \(c > 0\), we prove that, in the limit \(N \rightarrow \infty\), the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale \(M^{−1}\).