the early draft of the paper.

Preliminaries {#s.prelim}
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Dyson Brownian motion for covariance matrices
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Let $ B $ be an $ M \times N $ real matrix Brownian motion: $ B_{ij} $ are independent standard Brownian motions. We define the $ M \times N $ matrix $ M_t $ by $$M_t = M_0 + \dfrac{1}{\sqrt{N}}B_t.$$ The eigenvalues dynamics for the real Wishart process $ X_t := M_t^* M_t $ was first proved in [@bru1989diffusions]. Under our normalization convention, the equation is in the following form given in [@bourgade2017eigenvector Appendix C] $$\label{e.EigenvalueDBM}
{\mathrm{d}}\lambda_k = 2\sqrt{\lambda_k}\dfrac{{\mathrm{d}}B_{kk}}{\sqrt{N}} + \left( \dfrac{M}{N} + \dfrac{1}{N} \sum_{l \neq k}\dfrac{\lambda_k + \lambda_l}{\lambda_k - \lambda_l} \right){\mathrm{d}}t.$$ Due to technical issues, it is difficult to use the coupling method from [@bourgade2016fixed] to analyze in a direct way. This motivates us to consider the singular values instead.

Let $ s_k := \sqrt{\lambda_k} $ denote the singular values of $ X $. The Dyson Brownian motion for singular values dynamics of such sample covariance matrices is the following Ornstein-Uhlenbeck process [@erdHos2012local equation (5.8)]. $${\mathrm{d}}s_k = \dfrac{{\mathrm{d}}B_k}{\sqrt{N}} + \left[ -\dfrac{1}{2\xi}s_k + \dfrac{1}{2}\left( \dfrac{1}{\xi} -1 \right) \dfrac{1}{s_k} + \dfrac{1}{2N} \sum_{l \neq k} \left( \dfrac{1}{s_k - s_l} + \dfrac{1}{s_k + s_l} \right) \right]{\mathrm{d}}t, 