I have thought about this question for a long time and could only find partial answers.
The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian entries. For example, the eigenvalues are known to follow the Wigner semicircle distribution
$$ \mu_{N \times N} = \tfrac{1}{N}\sum \delta_{\lambda_i} \to \frac{2}{\pi}\sqrt{1 - x^2 }\, dx$$
In Queues and GUEs, Baryshnikov shows the eigenvalues of the minors the minors don't just interlace, if we condition on the spectrum of $H$ the eigenvalues are uniform in the Gelfand-Tsetlin polytope.
- Let $H$ be the random GUE matrix
- Let $H_k$ be the $k \times k$ upper left corner.
- The eigenvalues of $H_k$ and $H_{k+1}$ interlace.
- The eigenvalues of $H_1, \dots, H_N$ form a Gelfand-Tsetlin triangle
- Conditioning on the eigenvalues of $H$ the eigenvalues of the minors are uniform in the Gelfand-Tsetlin polytope.
Cedric Boutillier's discussion of says his bead model is a bi-infinite analogue of this uniform interlacing picture for GUE.
I am no longer sure it's possible to sample the eigenvalues of the GUE matrix as a uniform element of a polytope since the last row, the eigenvalues $\lambda$ of $H$ are distributed in a highly non-uniform way.
Is it possible to prove the Wigner semicircle law this way. Boaz Klartag writes a central limit theorem for high-dimensional convex sets.
Let $X$ be a random vector uniformly distributed in a convex polytope $K$ then $$ \left| \mathbb{P}[\langle X, \theta \rangle \in A] - \frac{1}{2\pi} \int_A e^{-t^2/2} dt\right| < \epsilon$$ where $X$ is centered $\mathbb{E}[X] = 0$ and isotropic $\mathrm{Cov}[X_i X_j] = \delta_{ij}$.
This point of view on large deviations could make sense since all the "stuff" for a high-dimensional polytopic is concentrated near the center. Is it possible to prove the Wigner semicircle formula along these lines?
