the entries of the $2\times 2$ operator converge pointwise, and are bounded above in an exponential way. This is what is done in the proof of Lemma 1 of @sosh, and we will show in **A8** that the arguments given there can be extended to handle our situation.

Further Remarks and Agenda
==========================

Most of the work in @imj is done in closed form, and in the finite dimensional case. That has two advantages from our standpoint: as the limiting behavior is only investigated in the last “step", most of the arguments given there carry through for our problem, and the method certainly does.

Therefore, our contribution is mostly technical; it follows very closely the ideas of @imj, providing solutions to technical problems appearing in the case we consider. Only at a few points could we not use the approach developed in @imj. This led us to an analysis of the complex case that is slightly different from the original one, but the core reasons for which the result holds are the same.

In what follows, we first focus on showing that (\[whattoshow1\]) and (\[whattoshow2\]) hold true when $n,N$ and their ratio tend to infinity. This takes care of the 