04-151 Rowan Killip, Irina Nenciu

Matrix Models for Circluar Ensembles (90K, Latex 2e) May 12, 04

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Abstract. The Gibbs distribution for $n$ particles of the Coulomb gas on the unit circle at inverse temperature $\beta$ is given by $$ \mathbb{E}^{\beta}_{n}(f)=\frac{1}{Z_{n,\beta}} \int\!\!\cdots\!\!\int\, f(e^{i\theta_1},\ldots,e^{i\theta_n})|\Delta(e^{i\theta_1},\ldots,e^{i\theta_n} )|^{\beta} \frac{d\theta_1}{2\pi}\cdots\frac{d\theta_n}{2\pi} $$ for any symmetric function $f$, where $\Delta$ denotes the Vandermonde determinant and $Z_{n,\beta}$ 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 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.

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