A Smooth Transition from Wishart to GOE

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.

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Fig. 1

Notes

  1. 1.

    In statistics the number of samples is usually denoted by n and the number of parameters is usually denoted by p, resulting in a \(p \times p\) Wishart matrix with n degrees of freedom. Here our notation is taken with the geometric perspective in mind, following [3,4,5].

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Acknowledgements

We thank an anonymous reviewer for helpful suggestions.

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Correspondence to Miklós Z. Rácz.

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Rácz, M.Z., Richey, J. A Smooth Transition from Wishart to GOE. J Theor Probab 32, 898–906 (2019). https://doi.org/10.1007/s10959-018-0808-2

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Keywords

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

Mathematics Subject Classification (2010)

  • 60B20