Eigenvalue density expansion in random matrices First, some background and context. Curently I am studying the basics of "random matrices" and their large N expansion. At large $N$ I know that the eigenvalues condense in what is called the Wigner semicircle. I see this function as the first term in a sort of asymptotic expansion. In fact I found this expansion
$$\rho(x)=\sum^\infty_{g=0}N^{-2g}\rho^{(g)}(x)+non-perturbative$$
it is equation (1.80) in

Random matrices. Bertrand Eynard, Taro Kimura, Sylvain Ribault arXiv:1510.04430.

That only even powers of $1/N$
appear in the perturbative terms is a mistery for me.
How we can prove that expansion? Can you give some references?
 A: In general the $1/N$ expansion does actually contain all powers, for example, in the Gaussian orthogonal ensemble (GOE, real symmetric matrices) the first two terms are
$$\rho(x)=\frac{2N}{\pi x_c^2}\sqrt{x_c^2-x^2}+\tfrac{1}{4}\delta(x-x_c)+\tfrac{1}{4}\delta(x+x_c)-\frac{1}{2\pi}(x_c^2-x^2)^{-1/2}+{\cal O}(N^{-1}),$$
where the order $N$ term is the Wigner semicircle, supported in $(-x_c,x_c)$.
For symplectic matrices (GSE) the expansion is similar, it is only for complex matrices (GUE) that the even powers of $1/N$ vanish.
When you do the $1/N$ expansion of $\rho(x)=\langle \sum_{i=1}^N\delta(x-x_n)\rangle$ for an eigenvalue distribution of the general form 
$$P(x_1,x_2,\ldots x_N)=\prod_{i<j}|x_i-x_j|^\beta\prod_k f(x_k),$$
you find that the order $N^0$ term is proportional to $2-\beta$ irrespective of the choice of $f(x)$. (So this is not something special for the Gaussian distribution.) In the GUE one has $\beta=2$, hence this term vanishes.
I don't know of an intuitive answer for this fact that does not require a calculation, but there may well be one.
