Global fluctuation formulas and universal correlations for random matrices and log-gas systems at infinite density

It is shown how the universal correlation function of Brezin and Zee, and Beenakker, for random matrix ensembles of Wigner-Dyson type with density support on a finite interval can be derived using a linear response argument and macroscopic electrostatics. The analogous formula for ensembles of unitary random matrices is derived using the same method, and the corresponding global fluctuation formula for the variance of a linear statistic is presented. The result for the universal correlation is checked using an exact result for the finite system two-point correlation in the Dyson circular ensemble at all even $\beta$.