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Local spectral statistics of the addition of random matrices

  • Ziliang CheEmail author
  • Benjamin Landon
Article
  • 5 Downloads

Abstract

We consider the local statistics of \(H = V^* X V + U^* Y U\) where V and U are independent Haar-distributed unitary matrices, and X and Y are deterministic real diagonal matrices. In the bulk, we prove that the gap statistics and correlation functions coincide with the GUE in the limit when the matrix size \(N \rightarrow \infty \) under mild assumptions on X and certain rigidity assumptions on Y (the latter being an assumption on the convergence of the eigenvalues of Y to the quantiles of its limiting spectral measure which we assume to have a density). Our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to Dyson Brownian motion. Our method also applies to the case when V and U are drawn from the orthogonal group. Our proof relies on the local law for H proved in Bao et al. (Commun Math Phys 349(3):947–990, 2017; J Funct Anal 271(3):672–719, 2016; Adv Math 319:251–291, 2017) as well as the DBM convergence results of Landon and Yau (Commun Math Phys 355(3):949–1000, 2017) and Landon et al. (Adv Math 346:1137–1332, 2019).

Mathematics Subject Classification

15B52 60B20 

Notes

Acknowledgements

The authors would like to thank H.-T. Yau and P. Sosoe for useful and enlightening discussions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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