, Volume 88, Issue 5-6, pp 1371-1386

Finite-N fluctuation formulas for random matrices

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Abstract

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic Σ j N =1 (x j − 〈x〉) is computed exactly and shown to satisfy a central limit theorem asN → ∞. For the circular random matrix ensemble the p.d.f.’s for the statistics ½Σ j N =1 (θ jπ) and − Σ j N =1 log 2 |sinθ j/2| are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem asN → ∞.

Communicated by J. L. Lebowitz