largest eigenvalue of $ Q $. For any $ {\varepsilon}>0 $, for large enough $ N $ we have $$\label{e.RateGeneralPopulation}
{\mathsf{d_K}}\left( \gamma_0 N^{2/3} (\mu_N - E_+),{\mathsf{TW}}\right) {\leqslant}N^{-\frac{1}{57} + {\varepsilon}},$$ where $ E_+ $ defined in denotes the rightmost endpoint of the spectrum and $ \gamma_0 $ is a normalization constant defined in .

The method of the paper follows the three-step strategy of the Erdös-Schlein-Yau dynamical approach [@erdHos2011universality]: (i) a priori bounds on locations of eigenvalues; (ii) local relaxation of the eigenvalue dynamics; (iii) a density argument showing eigenvalues statistics have not changed after short time.

Specifically, in this paper, the three-step strategy is employed in the following way: (i) is the rigidity for singular values, which can be rephrased from classical results on eigenvalues (see [@bloemendal2014isotropic; @pillai2014universality]). For the particular square case $ M \equiv N $, we use a different rigidity estimate at the hard edge from [@alt2017local],which results in a slightly different proof; (ii) is the recent approach to the analysis of Dyson Brownian motion from [@bourgade2018extreme], which introduced an observable defined via interpolation with integrable models (see [@bourgade2016fixed; @landon2019fixed]). It describes the singular values evolution through a stochastic advection equation; (iii) is a quantitative version of 