A Smooth Transition from Wishart to GOE

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.