the complex case problem - the matrix operator determinant can be computed as the product of two scalar operator determinants - and shows that the “reduced" variable he works with ought to have the same limit as it had in the Gaussian Orthogonal Ensemble case, which was studied in depth by Tracy and Widom.

For the sake of completeness, we recall that in this situation $\alpha=n-1-N$ and $$\mathbf{P}_N(l_1\leq t)=\sqrt{\det({\mathrm{Id}}+K_N\chi_t)} \;.$$ $K_N$ has the representation (in the $N$ even case) $$K_N=
\begin{pmatrix}
S_N+ \psi\otimes \epsilon \phi & S_N D-\psi\otimes\phi \\
{\epsilon}S_N -{\epsilon}+{\epsilon}\psi \otimes {\epsilon}\phi & S_N + {\epsilon}\phi \otimes \psi
\end{pmatrix} \; ,$$ where $D$ is the differential operator, ${\epsilon}$ is convolution with the kernel ${\epsilon}(x-y)$, and ${\epsilon}(x)=\text{sgn}(x)/2$. We note the slight change in $\alpha$ and replace $n$ by $n-1$ when we need to use the results or formulas derived in the complex case (for instance, the $S_N$ we just mentioned is $S_{n-1,N}$, and not $S_{n,N}$). We refer the reader to @ggk for a complement of information on operator determinants and to the end of section VIII in @tw96 for details on the technical problems that $K_N$ poses.

From a purely technical standpoint, one critical issue is to evaluate the large 