"Semiclassical approximation" in random matrix theory

Question

I am reading Planar Diagrams by Brezin, Parisi, Itzykson and Zuber. If you specialize the discussion in section 5 there we seem to derive the eigenvalue distribution of $N \times N$ random Hermitian matrices.

$$ N = \sum_k \theta(e_F - e_k) \approx \int \frac{d\lambda \, dp}{2\pi} \theta\bigg(e_F - \tfrac{1}{2}(p^2 + \lambda^2)\bigg) = \int \frac{d\lambda }{2\pi} \sqrt{2 e_F - \lambda^2 }\;\theta(2e_F - \lambda^2)$$

Here $\theta(x) = \mathbf{1}_{x \geq 0}$ is the Heaviside step function. Assuming this "semiclassical approximation" in the second equality, it looks like we have a very simple proof of the Wigner Semicircle Law just by setting Here I have set the constant $g = 0$ and $e_F$ is the "Fermi energy" (possibly zero).

The paper goes on to say the "large N vaccum diagrams" are equivalent to the ground state eigenvalues of the Laplacian $\Delta$ on the space of Hermitian matrices, under the Hilbert-Schmidt norm.

$$ H = -\tfrac{1}{2}\Delta + V = \bigg( \sum_k \tfrac{\partial^2}{\partial M_{kk}^2} + \sum_{i < j} \tfrac{\partial^2}{\partial M_{ij}^2} + \tfrac{\partial^2}{\partial \overline{M}_{ij}^2} \bigg) + \tfrac{1}{2}\mathrm{tr}M^2$$

This short proof seems a little too good to be true. What parts of that discussion are not rigorous? What is the mathematical counterpart of this semiclassical approximation and how is it justifified?

This post imported from StackExchange MathOverflow at 2014-12-01 12:41 (UTC), posted by SE-user john mangual