# Rates of convergence to Tracy-Widom?


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?

• I think it is an open question to design a Stein's method framework when the limiting distribution is a Tracy-Widom one. – user69642 May 26 '20 at 7:35

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 $${\cal O}[\min(n,p)]^{-2/3}$$.