\frac{1}{2\pi}\sqrt{(4-x^2)_+} $. By the relation , this means that the typical $ k $-th singular value $ \gamma_k $ of an $ N \times N $ data matrix shares the same position with the typical $ (N+k) $-th eigenvalue of a $ 2N \times 2N $ generalized Wigner matrix. The link between these two models can be illustrated by the symmetrization trick: Define the $ 2N \times 2N $ matrix $${\widetilde}{H} = \left(
\begin{matrix}
0 & X^*\\
X & 0
\end{matrix}
\right),$$ then we know that the eigenvalues of $ {\widetilde}{H} $ are precisely the symmetrized singular values of $ X^* $. Note that we have $ {\widetilde}{H}={\widetilde}{H}^* $, $ {\mathbb{E}}{\widetilde}{H}_{ij}=0 $ and $ \sum_{i=1}^{2N} {\mathbb{E}}{\widetilde}{H}_{ij}^2 = 1 $ for every $ j \in \llbracket 1,2N \rrbracket $. This shows that $ {\widetilde}{H} $ is indeed a Wigner-type matrix except the lack of nondegeneracy condition caused by the zero blocks. By considering the matrix of this type, the rigidity at the hard edge can be proved directly from [@alt2017local Theorem 2.7] $$\label{e.RigidityOneHard}
{\mathbb{P}}\left( |s_k - \gamma_k| {\leqslant}N^{-1+{\varepsilon}} \ \mbox{for all}\ k \in \llbracket 1,(1-\omega)N \rrbracket \right) > 1-N^{-D}.$$

Stochastic Advection Equation for Singular Values Dynamics {#s.LocalRelaxationFlow}
==========================================================

Stochastic advection equation
