We describe an ensemble of (sparse) random matrices whose eigenvalues
follow the Gibbs distribution
for n particles of the Coulomb gas on the
unit circle at inverse temperature beta. Our approach combines
elements from the theory of orthogonal polynomials on the unit circle with
ideas from recent work of
Dumitriu and Edelman. In particular, we resolve
a question left open by them: find a tri-diagonal model
for the Jacobi
ensemble
(Joint work with Rowan Killip. IMRN 50, 2004.)