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Something I really enjoy about Tao's writing is that he proves the same theorem over and over. While I complain a bit sometimes about clarity, this is a heuristic that I very much believe in.

This question is about the Wigner Semi-Circle Law from his 254A random matrix theory. He does it two different ways:

  • for a broad class of matrix ensembles Notes 4
  • for a Gaussian Ensembles Notes 6

They are Hermitan matrices: $M = \overline{M}^\dagger$ (equal to it's own adjoint) with elements $\xi_{ij}$ chosen under special conditions:

  • $\xi_{ij} = \overline{\xi_{ji}}$ is a complex variable, zero mean $\mathbb{E}[\xi_{ij}] = 0$ and unit variance $\mathbb{E}[\xi_{ij}^2] = 1$
  • Diagonal matrics $\xi_{ii} \in \mathbb{R}$ with bounded mean and variance.

So that semi-circle theorem works under very broad conditions, but let's set $\xi_{ij}$ and $\xi_{ii}$ be appropriate $\mathcal{N}(0,1)$ random variables. Before getting to the "main" proof in Notes 6, there are some rather unique and technical estimates that I'm going to skip, basically 90% of the proof.

He shows that GUE is a determinantal process with kernel basic on Hermite polynomials:

$$ K_n(x,x) = \sum_{j=0}^{n-1} \phi_j(x)^2 = \phi'_n(x)^2 + (n - x^2/4)\phi_n(x)^2 $$

with a related formula for $\mathbb{E}(\mu_{M_n/\sqrt{n}})$ . He explains to us the various notions of convergence for probability measures and how these interplay with matrix norms in general. And works a few of these cases out.

I recognize the Hermite polynomials (weighted with Gaussians) $$ L_h \phi = - h^2 \phi'' + \frac{x^2}{4} \phi = \left(- h^2 \frac{d^2}{dx^2} + \frac{x^2}{4} \right)\phi $$

Intuitively, I can picture what this "phase space" should look like. There should be a coordinate axis marked "$x$" and a momentum axis marked "$p$" and the Hamiltonian should read:

$$ L_h = p^2 + x^2/4 \leq 1 \text{ with }p = - i\hbar \frac{d}{dx} $$

which is indeed a circle (rather... an ellipse). There's some sort of projection operator: $$ \pi_{V_n} = L_{1/\sqrt{n}} \Big|_{[0,1]} =_{symbol} \dots (4-x^2)_+^{1/2}$$ Unfortunately, Tao warns us this requires semi-classical analysis or even microlocal analysis.

What does he mean by that? He does some type of "WKB approximation" except that's also not rigorous. So he settles for a mix of

  • variation of parameters
  • Gronwall's inequality
  • steepest descent

Google searching for "microlocal" analysis leads me astray. I found work of Kashiwara which I shall not even mention here...

How does WKB fail to be rigorous? What kind of machinery gets introduced with microlocal analysis?

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You need the theory of $h$-pseudodifferential operators.

Standard references are:

  1. "Spectral Asymptotics in the Semi-Classical Limit" by Dimassi and Sjöstrand.
  2. "An Introduction to Semiclassical and Microlocal Analysis" by Martinez.
  3. "Semiclassical analysis" by Zworski.
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