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In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix.

\[ \int e^{-\mathrm{Tr} \; M^2}dM \]

Ginibre calculated the density formula for the density of eigenvalues is symmetric in the variables:

\[ \rho(\vec{x}) = \prod_{i < j} \big|x_i - x_j\big|^2 e^{-\sum x_i^2} = \det K_n(x_i, x_j) \]

Gaudin and Mehta calculated the kernel for this eigenvalue distribition to be a sum over Hermite polynomials:

\[ K_n(x,y) = \sum_{k=0}^{n-1} \phi_k(x) \phi_k(y) = e^{-x^2} \sum_{k=0}^{n-1} P_k(x) P_k(y) \]

The functiosn $\phi_k$ are eigenfunctions of the Harmonic oscillator:

\[ H \; \phi = \left( -\frac{d^2}{dx^2} + \frac{x^2}{4}\right)\phi = E \; \phi \hspace{0.25in}\text{ so that }\hspace{0.25in} H\phi_k = \left(k + \frac{1}{2} \right)\phi_k\]

Semiclassical ( WKB ) approximation

Remark #1 in those notes indicate this is a Schrodinger operator and a projection operator in phase space.

\[ H = p^2 + \frac{x^2}{4} \leq n + \frac{1}{2} \]

The energy levels correspond to circles or ellipses. We have $n$ fermions restricted to this region of phase space. Rescale by the number of fermions and project to x-space: the semicircle distribution emerges.

You can already see this Fermi gas picture in Section 5 of the 1978 paper by Brezin Parisi Itzykson and Zuber.


The semiclassical approach can save space in certain discussions and it's use in statistical mechanics of harmonic oscillators (e.g.)

I would like to derive the semicircle law this way, although Tao's disclaimer scares me a little bit:

For sake of completeness, I am recording some notes on this approach here, although to focus on the main ideas I will not be completely rigorous in the derivation (ignoring issues such as convegence of integrals or of operators, or (removable) singularities in kernels caused by zeroes in the denominator).

Where do the WKB approximation & the Fermi-Gas picture fail to be rigorous in this case?
While, these discussions certainly have gaps, perhaps no one will be the wiser.

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The semicircle theorem itself is addressed in section 4 of Tao's notes terrytao.wordpress.com/2010/02/23/… He outlines careful approaches through WKB approximation and through steepest descent. –  john mangual Jul 2 '13 at 19:55
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