Matrix models for circular ensembles

Abstract

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.

28 pages

Cited by in corpus (16)

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• Statistics of extremes in eigenvalue-counting staircases
• The bead process for beta ensembles
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• Asymptotic forms for hard and soft edge general $\beta$ conditional gap probabilities
• Convergence and an explicit formula for the joint moments of the Circular Jacobi $\beta$-Ensemble characteristic polynomial
• Uniform point variance bounds in classical beta ensembles
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