Matrix models for circular ensembles

The Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature β is given by $\text{E}_n^\beta(f)=(1/Z_{n,\beta})\int\cdots\int f(e^{i\theta_1},\dots,e^{i\theta_n}){\vert\Delta(e^{i\theta_1},\dots,e^{i\theta_n})\:\vert}^\beta(d\theta_1/2\pi)\dots d\theta_n/2\pi$ for any symmetric function f, where Δ denotes the Vandermonde determinant and Zn,β the normalization constant. We will describe an ensemble of (sparse) random matrices whose eigenvalues follow this distribution. Our approach combines elements from the theory of orthogonal polynomials on the unit circle with ideas from a recent work of Dumitriu and Edelman. In particular, we resolve a question left open by them: finding a tridiagonal model for the Jacobi ensemble.