m_{{\mathsf{MP}}}(z^2) \right|.$$ Combining with the strong local Marchenko-Pastur law, this gives us $$\label{e.LocalLaw}
{\mathbb{P}}\left( \left| m_N(z) - m(z) \right| \lesssim \dfrac{\varphi}{N\eta} \right) > 1-N^{-D}.$$

![Graphs of the density $ \rho(x) $ for $ x{\geqslant}0 $ (i.e. the density for the actual singular values) with $ \xi=1,(3/5)^2,(3/10)^2 $ respectively. The curve for $ \xi=1 $ does not have a “square root”-type shape at the edge $ \sqrt{\lambda_-} =0 $ in this case due to the singularity of the Marchenko-Pastur distribution.](density.jpg){height="4cm"}

For the rigidity estimates, a key observation is that the critical case $ \xi=1 $ is significantly different from other cases. This is because the Marchenko-Pastur law $ \rho_{{\mathsf{MP}}} $ has a singularity at the point $ x=0 $ in this situation. When $ \xi < 1 $, the rigidity of singular values can be easily obtained from the analogous estimates for eigenvalues (see [@pillai2014universality]). Let $ {\widehat}{k}:=\min(k,N+1-k) $, for any $ D>0 $ there exists $ N_0(D) $ such that the following holds for any $ N {\geqslant}N_0 $, $$\label{e.RigidityNotOne}
{\mathbb{P}}\left(|s_k - \gamma_k| {\leqslant}\varphi^{\frac{1}{2}} N^{-\frac{2}{3}}({\widehat}{k})^{-\frac{1}{3}} \  \mbox{for all}\ k \in \llbracket 1,N \rrbracket \right) > 1-N^{-D}.$$

For the critical case $ \xi=1 $, now the Marchenko-Pastur distribution is supported on 