on $n$ and $N$, but as in @imj, we choose to not carry these indices in the interest of readability. Also, $\phi$ and $\psi$ admit more “compact" representations, in terms of a single Laguerre polynomial, with a modified $\alpha$, or another degree. These are easy to derive using @szego, p.102, for instance. Nevertheless we choose to work (except in **A7**) with the previous representations because of the symmetries they present.

The kernel $S_N$ mentioned in (\[freddet\]) has the representation (@imj, equation (3.6)) $$S_N(x,y)=\int_{0}^{\infty} \phi(x+z)\psi(y+z)+ \psi(x+z)\phi(y+z) dz \; .$$

Now let $\bar{S}$ be the Airy operator. Its kernel is $$\bar{S}(x,y)=\frac{{\mathrm{Ai}}(x){\mathrm{Ai}}'(y)-{\mathrm{Ai}}(y){\mathrm{Ai}}'(x)}{x-y}=\int_0^{\infty}{\mathrm{Ai}}(x+u){\mathrm{Ai}}(y+u) du \; ,$$ where Ai denotes the Airy function. It was shown in @tw94 that, viewing $\bar{S}$ as an operator on $L^{2}[s,\infty)$, one had $$\det({\mathrm{Id}}-\bar{S})=W_2(s) \; ,$$ where $W_2$ is the Tracy-Widom law “emerging" in the complex case (see **A0**). So the complex analog of theorem \[ThJ\] follows from the fact that, after defining $S_{\tau}(x,y)={\sigma_N}S_N(\mu_N+{\sigma_N}x,\mu_N+{\sigma_N}y)$, Johnstone managed to show, for all $s$, that $$\det({\mathrm{Id}}-S_{\tau}) {\rightarrow}\det({\mathrm{Id}}-\bar{S}) \; .$$

To do this, he introduced $\phi_{\tau}(s)=\sigma_N \phi(\mu_N+s\sigma_N)$, and similarly $\psi_{\tau}$. Note that we have $$S_{\tau}(x,y)=\int_{0}^{\infty} \phi_{\tau}(x+z)\psi_{\tau}(y+z)+ \psi_{\tau}(x+z)\phi_{\tau}(y+z) dz \; .$$ Since what we are interested in is really $S_{\tau} \chi_s$, for some fixed $s$, 