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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)

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