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$\renewcommand{\!}{\mathbf} \renewcommand{\Ai}{\operatorname{Ai}}$ One can define the Tracy-Widom distribution as the Fredholm determinant $F_2(t)=\det(\mathbf I-\mathbf A)$ where $$\mathbf A(x, y)=\begin{cases} \frac{\Ai(x) \Ai'(y)-\Ai'(x) \Ai(y)}{x-y} & \text {if } x \neq y \\ \Ai'(x)^{2}-x \Ai(x)^{2} & \text {if } x=y \end{cases} \text{ for } \Ai(x) = \frac 1{\pi} \int_0^\infty \cos\Big(\tfrac 13 t^3 + xt\Big) \, d t$$ It's well known that the largest eigenvalue of an $n\times n$ GUE matrix (appropriately scaled) converges in distribution to $F_2$. Also, by the celebrated Baik-Deft-Johansson theorem, the length of the longest increasing subsequence of a random permutation $\in S_n$ (appropriately scaled) also converges in distribution to $F_2$. I'm sure there are many other examples of situations in which $F_2$ appears as the limiting distribution.

However, I have been trying to find references about how fast the things that converge to $F_2$ converge to $F_2$. My searches have turned up this paper: https://arxiv.org/pdf/0803.3408.pdf and this paper: https://arxiv.org/pdf/1901.05235.pdf, both of which seem to answer related questions but not exactly what I'm looking for.

Does anyone know the rates at which the largest eigenvalue of a GUE matrix and/or the longest increasing subsequence of a random permutation converge to $F_2$? If not exactly, are there references that have done numerical computations and put forward a conjecture about such rates?

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    $\begingroup$ I think it is an open question to design a Stein's method framework when the limiting distribution is a Tracy-Widom one. $\endgroup$ – user69642 May 26 at 7:35
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Accuracy of the Tracy-Widom limits for the extreme eigenvalues in white Wishart matrices studies the rate of convergence for Wishart matrices $WW^T$, with $W$ an $n\times p$ matrix with i.i.d. Gaussian matrix elements. The error is ${\cal O}[\min(n,p)]^{-2/3}$.

Here is an error plot, showing for two percentiles of the exact distribution the relative error of the Wishart distribution. The errors are smaller further out in the tail of the distribution.

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