mentioned in Section \[s.LocalLaw\], discussions for the case $ \xi=1 $ are expected to be different due to the singularity of the Marchenko-Pastur distribution (which results in a distinct shape of the density $ \rho(x) $ for singular values). Therefore, in this and the subsequent section, we first discuss the case $ \xi \neq 1 $ and will show how to adapt the proof to the case for square data matrices in Section \[s.CaseOne\].

Denote $$\kappa(z) := \min\left\{\left|z-\sqrt{\lambda_-}\right|,\left|z-\sqrt{\lambda_+}\right|\right\},$$ and $$a(z) := \operatorname{dist}\left(z,\left[\sqrt{\lambda_-},\sqrt{\lambda_+}\right]\right),\ \ \ b(z):= \operatorname{dist}\left( z, \left[\sqrt{\lambda_-},\sqrt{\lambda_+}\right]^c \right)$$ We consider the curve $${\mathscr{S}}:= \left\{ z=E+{\text{\rm i}}y: \sqrt{\lambda_-} + \varphi^2 N^{-2/3} <E< \sqrt{\lambda_+} - \varphi^2 N^{-2/3},\ y=\varphi^2/\left( N \kappa(E)^{1/2} \right) \right\},$$ and the domain $ \mathscr{R} := \cup_{0<t<1}\{z_t:z \in {\mathscr{S}}\} $.

\[l.characteristics\] Uniformly in $ 0<t<1 $ and $ z=z_0 $ satisfying $ \eta := {\text{\rm Im}\hspace{0.1cm}}z>0 $ and $ |z-\sqrt{\lambda_+}|<\sqrt{\xi}/10 $, we have $${\text{\rm Re}\hspace{0.1cm}}(z_t -z_0) \sim t \dfrac{a(z)}{\kappa(z)^{1/2}}+t^2,\ \ \ \ {\text{\rm Im}\hspace{0.1cm}}(z_t-z_0) \sim t\dfrac{b(z)}{\kappa(z)^{1/2}}.$$ In particular, if in addition we have $ z \in {\mathscr{S}}$, then $$(z_t - z_0) \sim \left( t \dfrac{\varphi^2}{N \kappa(E)} +t^2 \right) + {\text{\rm i}}\kappa(E)^\frac{1}{2} t.$$ Moreover, for any $ \kappa >0 $, uniformly in $ 0<t<1 $ and $ z=E+{\text{\rm i}}\eta \in 