(i.e the real and imaginary parts are independent, and they are both ${\cal N}(0,1/2)$), rather than with entries that are ${\cal N}(0,1)$. When we mention the complex case, we refer to this situation.

We now give a quick overview of the important points around which the proof of Theorem \[ThJ\] was articulated.

At the core of several random matrix theory results lie the fact that the joint distribution of the eigenvalues of the random matrices of interest is known and can be represented as the Fredholm determinant of a certain operator (or a totally explicit function of it).

Building on this, if we introduce a number $b$ that is $1$ in the real case and $2$ in the complex one, it turns out that one has the representation formula $$\label{freddet}
{\mathbf{E}_{N}\left(\prod_{i=1}^N (1+f(l_i))\right)}=\left[\det({\mathrm{Id}}+S_Nf)\right]^{b/2} \; ,$$ where $S_N$ is an explicit kernel, depending of course upon the kind of matrices in which one is interested. Here, $f$ treated as an operator means multiplication by this function. It is clear that if $\chi_t=-\mathbf{1}\{x:x\geq t\}$, we have $$\mathbf{P}_N(l_1\leq t)=\left[\det({\mathrm{Id}}+S_N\chi_t)\right]^{b/2}\;.$$

The interested reader can find background information on this in @mehta, chapters 5 and 6, @tw98 or @deift, chapter 5, which in turn (p.109) points 