of $\mu_N$ ($={1 \over 2}+{1 \over N}$ by eq.(\[Ord\])); hence $\mu$ behaves regularly (as opposed to the later examples).

As a crude dividing line between (generic) singular and regular behaviours, we expect that essentially those quantities which converge both for finite $N$ [*and*]{} in the limiting problem should be regular, in particular: $$\label{REG}
Z_N^\pm(s,\lambda) \to Z_\infty^\pm (s,\lambda) \quad 
\mbox{ iff } \Re (s) > \mu_\infty .$$ Singular behaviour should then set in at $\Re (s) = \mu_\infty$ and, plausibly, become worse as $\Re (s)$ decreases further. This purely qualitative argument cannot however predict the precise behaviour of any singular quantity.

Fortunately, quantitative statements are made easier here by a set of explicit results for the finite-$N$ problems (cf. Appendix A.1) and their counterparts for $N=+\infty$ (Sec.2.1 above). Those data indeed behave consistently with the prediction (\[REG\]) taken with $\mu_\infty = {1 \over 2}$. As examples of regular ($s>\mu_\infty$) behaviours, $Z_N^\pm(1) \to Z_\infty^\pm(1) = {1/2 \atop 1/6}$ , and likewise for the higher-order [*sum rules*]{} involving $s=2,\ 3,\cdots$ (Sec.2.5 below). As opposite examples (involving $s <\mu_\infty$): $Z_N(0) \equiv 0$ for all $N$ whereas $Z_\infty (0)=-1/2$ by eqs.(\[Z0\],\[Zi0\]); and worse, eq.(\[D0\]) implies $$\label{D0s}
\exp [-( Z_N^\pm)' (s=0)] = D_N^\pm(\lambda=0) \sim (N/\pi)^{1/2} \to 