The Kolmogorov distance from the rightmost Ginibre eigenvalue to Gumbel is exactly 25 log log n/(4e log n) and the W1 distance is exactly 25 log log n/(4 log n), with analogous rates for the spectral radius.
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Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution
The Kolmogorov distance from the rightmost Ginibre eigenvalue to Gumbel is exactly 25 log log n/(4e log n) and the W1 distance is exactly 25 log log n/(4 log n), with analogous rates for the spectral radius.