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Random matrix ensembles with an effective extensive external charge. (English) Zbl 0912.15030
Studies of chaotic scattering have encountered ensembles of random matrices in which the eigenvalue probability density function contains a one-body factor with an exponent proportional to the number of eigenvalues. Two such ensembles have been encountered; Poisson kernel, and the Laguerre ensemble. The authors consider various properties of these ensembles. Jack polynomial theory is used to prove a reproducing property of the Poisson kernel, and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the Dyson circular ensemble. For the Laguerre ensemble, the scaled global density is calculated exactly for all even values of the parameter \(\beta\), while for \(\beta=2\), the neighborhood of the smallest eigenvalue is shown to be in the soft edge universality class.
The paper contains the sections of Introduction, the Poisson kernel, Laguerre ensemble with an \(N\)-dependent exponent, the distribution functions in the neighborhood of the smallest eigenvalue.
Reviewer: Y.Kuo (Knoxville)

15B52 Random matrices (algebraic aspects)
60H25 Random operators and equations (aspects of stochastic analysis)
15A18 Eigenvalues, singular values, and eigenvectors
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