arXiv subjects

Alex Bloemendal

Publications and source records attributed to Alex Bloemendal.

4 recordsLinked to original sources

arXiv2016

Limits of spiked random matrices II

The top eigenvalues of rank $r$ spiked real Wishart matrices and additively perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition in the large size limit. We show that they have limiting distributions for near-critical perturbations, fully resolving the conjecture of Baik, Ben Arous and Péché [Duke Math. J. (2006) 133 205-235]. The starting point is a new $(2r+1)$-diagonal form that is algebraically natural to the problem; for both models it converges to a certain random Schrödinger operator on the half-line with $r\times r$ matrix-valued potential. The perturbation determines the boundary condition and the low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. We treat the real, complex and quaternion ($β=1,2,4$) cases simultaneously. We further characterize the limit laws in terms of a diffusion related to Dyson's Brownian motion, or alternatively a linear parabolic PDE; here $β$ appears simply as a parameter. At $β=2$, the PDE appears to reconcile with known Painlevé formulas for these $r$-parameter deformations of the GUE Tracy-Widom law.

math.PR↗

arXiv2015

Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices

We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur 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 $\langle v, (X^* X - z)^{-1} w \rangle - \langle v,w\rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v, w \in \mathbb C^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $Im z \geq N^{-1+ε}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.

math.PR↗

arXiv2015

On the principal components of sample covariance matrices

We introduce a class of $M \times M$ sample covariance matrices $\mathcal Q$ which subsumes and generalizes several previous models. The associated population covariance matrix $Σ= \mathbb E \cal Q$ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of $Σ- I_M$ may depend on $M$ in an arbitrary fashion. We investigate the principal components, i.e.\ the top eigenvalues and eigenvectors, of $\mathcal Q$. We derive precise large deviation estimates on the generalized components $\langle \mathbf w, \boldsymbol ξ_i \rangle$ of the outlier and non-outlier eigenvectors $\boldsymbol ξ_i$. 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 \emph{subcritical} spikes of $Σ$. The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix $\mathcal Q$, satisfying $\mathbb E \mathcal Q = I_M$, as input: the isotropic local Marchenko-Pastur law established in [9], 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.

math.PR↗

arXiv2011

Limits of spiked random matrices I

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 beta analogue, proving a conjecture of Baik, Ben Arous and Péché (2005). We also treat shifted mean Gaussian orthogonal and beta 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 beta appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at beta=2,4, yielding in particular a new and simple proof of the Painlevé representations for these Tracy-Widom distributions.

math.PR↗