Matrix models for circular ensembles

2004/10/02 by R. Killip, I. Nenciu · 3 citations
Mathematics · Physics and Astronomy · #math.SP #math-ph #math.MP #msc:15A52

paper · pdf

published as IMRN 50 (2004), 2665--2701 · 28 pages

arxiv created 2004/10/02 · arxiv updated 2009/12/01

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.

Cited by

• The smallest eigenvalue of β-Laguerre and β-Jacobi ensembles and multivariate orthogonal polynomials
• Sum rules via large deviations: polynomial potentials and multi-cut regime on the unit circle
• Debiasing Polynomial and Fourier Regression

Related

• Random normal matrices in the almost-circular regime
• Circular Rosenzweig-Porter random matrix ensemble
• Limits for circular Jacobi beta-ensembles
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• Random matrices, random polynomials and Coulomb systems