the leading order contribution as $$\begin{aligned}
\label{LargeNdx}
\langle\,(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle
&=& 
\dfrac{\sigma^{2}}{2N}
\Bigg\{\dfrac{5}{2}+\dfrac{1}{8}\,
\dfrac{
\left(
|\Omega|^{2}-|\Omega|^{-2}
\right)}
{\text{ln}(|\Omega|)}
+
g(k_{\alpha})\Bigg\},
\\
\nonumber
\\
\label{LargeNdy}
\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle
&=&
-\dfrac{\sigma^{2}}{2N}
\Bigg\{\dfrac{1}{2}-\dfrac{1}{8}\,
\dfrac{
\left(
|\Omega|^{2}-|\Omega|^{-2}
\right)}
{\text{ln}(|\Omega|)}
+
g(k_{\alpha})\Bigg\},\end{aligned}$$ where $$\begin{aligned}
g(k_{\alpha})&=&
\dfrac{1}{8}
\left(
\dfrac{e^{-2ik_{\alpha}}\Omega^{2}-e^{2ik_{\alpha}}\Omega^{-2}}
{4i \sin(k_{\alpha})\left(2N\cos(k_{\alpha})+i\sin(k_{\alpha})\text{ln}(\Omega)\right)}
\right) 
+
(\text{c.c.}).\end{aligned}$$

Equations (\[abdz7\], \[LargeNdx\], \[LargeNdy\]) are our main analytical results and they are also valid for parametric resonances with the $\Omega$ given in Eq. (\[Omgpara\]). In Fig. \[ex\_finitevsLarge\] we verify the asymptotic results (\[abdz7\], \[LargeNdx\], \[LargeNdy\]) against their finite-$N$ counterparts, for exact resonances with $N=100$ and $\eta=0.81$. Fig. \[pa\_finitevsLarge\] is repeated on the same pattern but for parametric resonances where $N=500$ and $\eta=0.81$. They confirm that the asymptotic results give a good account for the finite-$N$ results. However, there are some exception near the edges (not visible on the scale of the plot) where the ansatz (\[ansatz\]) is not valid.

It turns out that in order to calculate the DOR we need the second order corrections $(\delta_{2}\tilde{\mathcal{Z}}_{\alpha})$, derived in Eq. (\[dz2\]). We have followed the method used earlier [@FZ:99] for Hatano-Nelson Model [@HN:97]. However, we have not been able to obtain a closed expression of the DOR. This is discussed in Appendix B where we 