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 transitionMathematics Subject Classification (2010)
60B20Notes
Acknowledgements
We thank an anonymous reviewer for helpful suggestions.
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