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?


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