Derivation of an asymptotic expression in Beenakker’s general fluctuation formula for random-matrix correlations near an edge

We use a linear response argument and macroscopic electrostatics to derive the formula 〈σ(x)σ(x’)${\mathrm{〉}}^{\mathit{T}}$∼-(x+x’) /[β2${\mathrm{\pi }}^2$(xx’ ${)}^{1/2}$(x-x’ ${)}^2$], where σ(x) denotes the charge density, for general logarithmic-potential Coulomb systems on a half-line x>0.