A Smooth Transition from Wishart to GOE

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Abstract

It is well known that an \(n \times n\) Wishart matrix with d degrees of freedom is close to the appropriately centered and scaled Gaussian orthogonal ensemble (GOE) if d is large enough. Recent work of Bubeck, Ding, Eldan, and Racz, and independently Jiang and Li, shows that the transition happens when \(d = \Theta ( n^{3} )\). Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when \(d / n^{3} \rightarrow c \in (0, \infty )\). This shows, in particular, that the phase transition from Wishart to GOE is smooth.

Keywords

Random matrix theory Wishart distribution Gaussian Orthogonal Ensemble (GOE) Total variation Phase transition 

Mathematics Subject Classification (2010)

60B20 

Notes

Acknowledgements

We thank an anonymous reviewer for helpful suggestions.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Microsoft ResearchRedmondUSA
  2. 2.University of WashingtonSeattleUSA

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