Anisotropic local laws for random matrices

By Antti Knowles, Jun Yin. 2014

224 citations recorded by the source corpus.

We develop a new method for deriving local laws for a large class of random matrices. It is applicable to many matrix models built from sums and products of deterministic or independent random matrices. In particular, it may be used to obtain local laws for matrix ensembles that are anisotropic in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity. For definiteness, we present the method for sample covariance matrices of the form , where T is deterministic and X is random with independent entries. We prove that with high probability the resolvent of Q is close to a deterministic matrix, with an optimal error bound and down to optimal spectral scales. As an application, we prove the edge universality of Q by establishing the Tracy–Widom–Airy statistics of the eigenvalues of Q near the soft edges. This result applies in the single-cut and multi-cut cases. Further applications include the distribution of the eigenvectors and an analysis of the outliers and BBP-type phase transitions in finite-rank deformations; they will appear elsewhere. We also apply our method to Wigner matrices whose entries have arbitrary expectation, i.e. we consider $$W+A$$W+A where W is a Wigner matrix and A a Hermitian deterministic matrix. We prove the anisotropic local law for $$W+A$$W+A and use it to establish edge universality.

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

View Paper: https://www.semanticscholar.org/paper/a3eafa3b2eee84b110c5699567320d335567f937

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Citations in the corpus

Cleaning large correlation matrices: tools from random matrix theory (2016, background): “Note that the results presented above only cover a small part of the extremely vast literature on this topic, including the study microscopic/local statistics (down to the N−1 scale) [97,98,113,124].”
Surprises in High-Dimensional Ridgeless Least Squares Interpolation (2019, background): “Technically, [57, 69] apply asymptotic random matrix theory results, such as [43], while we have to take a longer detour to exploit nonasymptotic results established in [41].”
Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, methodology): “2, we introduced the classes of parameters HP(cn,p,M) defined in (8) to model the sparse structure of precision matrices. We have seen that Sun and Zhang (2013) show that the estimator Ω̃ attains the rate of convergence c(2)n,p(log p)/n under the spectral norm over HP(cn,p,M) in Theorem 13 of Sectio”
A Dynamical Approach to Random Matrix Theory (2017, unknown): “The edge universality was achieved in [89,94].”
Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, methodology): “ R, where E + ≡ E +(N) is given in (2.13) and γ 0 = γ 0(N) is given in (2.17). 6 Remark 2.8. For non-Gaussian X and general oﬀ-diagonal Σ, we can combine our results with edge universality results in [6, 30] to identify the Tracy-Widom distribution for the largest eigenvalues. 2.4. Applications. In ”
Singular vector and singular subspace distribution for the matrix denoising model (2018, methodology): “The key ingredient in our estimation is a special case of the anisotropic local law derived in [43], which is essentially the isotropic local law previously derived in [12].”
A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices (2016, methodology): “In fact under different assumptions, they have been proved previously in different forms [6, 31].”