the case when $M$ sites are connected to the outer world where $1\le M\le N$ [@Borgonovi:2012]. We believe that our methods could be useful for the study of such models. Finally, we mention the case where $\xi(0)\sim\mathcal{O}(N)$. It requires a separate investigation as our perturbative analysis fails in this limit.

The author is thankful to Boris Shapiro for suggesting the problem to him. The author would also like to give credit to Joshua Feinberg for the derivation of some of the equations in Secs. III and IV and also for the help in Appendix B. Discussions with both of them are gratefully acknowledged. The author also acknowledges Marko Žnidarič and Thomas H. Seligman for reading the manuscript.

Support from ISF-1067 and generous hospitality of Technion Institute are also acknowledged. Additional support by the project 79613 by CONACyT, Mexico, is acknowledged.

Large-$N$ behavior of the denominator in (\[FPT2\])
===================================================

The denominator in Eq. (\[FPT2\]) can be simplified as follows: $$\begin{aligned}
\label{app1}
D(z_{\alpha})
&=&
-N+U_{N-1}(z_{\alpha}/2)T_{N+1}(z_{\alpha}/2)-i\eta\exp[I(z_{\alpha})]
\nonumber
\\
&\times&
\sin[k(z_{\alpha})][U_{N-1}(z_{\alpha}/2)]^{2}
\nonumber
\\
&\approx&
-N+\Big[-\exp(-ik_{\alpha})[\Omega(1-\eta)+1]
\nonumber
\\
&+&
\exp(ik_{\alpha})[\Omega^{-1}(1+\eta\exp(2ik_{\alpha}))+1-2\eta]
\Big]
\nonumber
\\
&\times&
\Big[2[\exp(-ik_{\alpha})-\exp(ik_{\alpha})]\Big]^{-1}.\end{aligned}$$ Using now $$\Omega(1-\eta)+1=2-\eta \exp(2ik_{\alpha}),$$ and $$\Omega^{-1}(1+\eta\exp(2ik_{\alpha}))+1-2\eta
=
\dfrac{2-3\eta+\eta^{2}\exp(2ik_{\alpha})}
{1-\eta\exp(2ik_{\alpha})},
\nonumber
\\$$ in (\[app1\]) we get $$\begin{aligned}
D
&\simeq& 
-N-\dfrac{1}{1-\eta\exp(2ik_{\alpha})}.\end{aligned}$$ For 