# Local laws for multiplication of random matrices

Ding X, Ji HC. 2023. Local laws for multiplication of random matrices. The Annals of Applied Probability. 33(4), 2981–3009.

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https://doi.org/10.48550/arXiv.2010.16083
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*Journal Article*|

*Published*|

*English*

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Author

Ding, Xiucai;
Ji, Hong Chang

^{ISTA}Department

Abstract

Consider the random matrix model A1/2UBU∗A1/2, where A and B are two N × N deterministic matrices and U is either an N × N Haar unitary or orthogonal random matrix. It is well known that on the macroscopic scale (Invent. Math. 104 (1991) 201–220), the limiting empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution
of the limiting ESDs of A and B, denoted as μα μβ, where μα and μβ are the limiting ESDs of A and B, respectively. In this paper, we study the asymptotic microscopic behavior of the edge eigenvalues and eigenvectors statistics. We prove that both the density of μA μB, where μA and μB are the ESDs of A and B, respectively and the associated subordination functions
have a regular behavior near the edges. Moreover, we establish the local laws near the edges on the optimal scale. In particular, we prove that the entries of the resolvent are close to some functionals depending only on the eigenvalues of A, B and the subordination functions with optimal convergence rates. Our proofs and calculations are based on the techniques developed for the additive model A+UBU∗ in (J. Funct. Anal. 271 (2016) 672–719; Comm. Math.
Phys. 349 (2017) 947–990; Adv. Math. 319 (2017) 251–291; J. Funct. Anal. 279 (2020) 108639), and our results can be regarded as the counterparts of (J. Funct. Anal. 279 (2020) 108639) for the multiplicative model.

Publishing Year

Date Published

2023-08-01

Journal Title

The Annals of Applied Probability

Acknowledgement

The first author is partially supported by NSF Grant DMS-2113489 and grateful for the AMS-SIMONS travel grant (2020–2023). The second author is supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.
The authors would like to thank the Editor, Associate Editor and an anonymous referee for their many critical suggestions which have significantly improved the paper. We also want to thank Zhigang Bao and Ji Oon Lee for many helpful discussions and comments.

Volume

33

Issue

4

Page

2981-3009

ISSN

IST-REx-ID

### Cite this

Ding X, Ji HC. Local laws for multiplication of random matrices.

*The Annals of Applied Probability*. 2023;33(4):2981-3009. doi:10.1214/22-aap1882Ding, X., & Ji, H. C. (2023). Local laws for multiplication of random matrices.

*The Annals of Applied Probability*. Institute of Mathematical Statistics. https://doi.org/10.1214/22-aap1882Ding, Xiucai, and Hong Chang Ji. “Local Laws for Multiplication of Random Matrices.”

*The Annals of Applied Probability*. Institute of Mathematical Statistics, 2023. https://doi.org/10.1214/22-aap1882.X. Ding and H. C. Ji, “Local laws for multiplication of random matrices,”

*The Annals of Applied Probability*, vol. 33, no. 4. Institute of Mathematical Statistics, pp. 2981–3009, 2023.Ding, Xiucai, and Hong Chang Ji. “Local Laws for Multiplication of Random Matrices.”

*The Annals of Applied Probability*, vol. 33, no. 4, Institute of Mathematical Statistics, 2023, pp. 2981–3009, doi:10.1214/22-aap1882.**All files available under the following license(s):**

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arXiv 2010.16083