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 $d / n^{3} to c in (0, infty)$. This shows, in particular, that the phase transition from Wishart to GOE is smooth.