For sparse random matrices with q ≥ N^{1/6+δ}, the largest eigenvalue, after a small model-dependent correction, converges to Tracy-Widom at rate almost N^{-1/3} + N^{2/3}/q^4.
Convergence rate to the Tracy-Widom laws for the largest eigenvalue of Wigner matrices
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Quantitative Tracy-Widom laws for sparse random matrices
For sparse random matrices with q ≥ N^{1/6+δ}, the largest eigenvalue, after a small model-dependent correction, converges to Tracy-Widom at rate almost N^{-1/3} + N^{2/3}/q^4.