A Smooth Transition from Wishart to GOE

Miklós Z. Rácz () and Jacob Richey ()
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Miklós Z. Rácz: Microsoft Research
Jacob Richey: University of Washington

Journal of Theoretical Probability, 2019, vol. 32, issue 2, 898-906

Abstract: Abstract It is well known that an $$n \times n$$ n × 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 $$d = \Theta ( n^{3} )$$ d = Θ ( n 3 ) . Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when $$d / n^{3} \rightarrow c \in (0, \infty )$$ d / n 3 → c ∈ ( 0 , ∞ ) . 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; 60B20 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-018-0808-2

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