$ [0,4] $ and is given by $ \rho_{{\mathsf{MP}}}(x)= \frac{1}{2\pi}\sqrt{(4-x)/x} $. A key observation is that the scales of eigenvalue spacings are different at the two edges. Due to this phenomenon, we use the following two different results, depending on the location in the spectrum.

On the one hand, the Marchenko-Pastur distribution still behaves like a square root near the soft edge $ x=4 $, which implies that the result is the same as the rectangular case. The rigidity estimate near the soft edge can be easily adapted from the result for eigenvalues in [@bloemendal2014isotropic Theorem 2.10], i.e. for some (small) $ \omega>0 $ and any $ {\varepsilon}>0 $ we have $$\label{e.RigidityOneSoft}
{\mathbb{P}}\left( |s_k - \gamma_k| {\leqslant}N^{\varepsilon}(N-k+1)^{-\frac{1}{3}}N^{-\frac{2}{3}} \ \mbox{for all} \ k \in \llbracket (1-\omega)N,N \rrbracket \right) > 1-N^{-D}.$$

On the other hand, as explained in [@cacciapuoti2013local], at the hard edge $ x=0 $ the typical distance between eigenvalues and the edge is of order $ N^{-2} $, which is much smaller than the typical distance between neighbouring eigenvalues in the bulk (or at the soft edge). Note that in this situation, the measure for the symmetrized singular values coincides with the standard semicircle law, that is $ \rho(x) = 