Amplitude Function of Asymptotic Correlations Along Charged Wall in Coulomb Fluids

In classical semi-infinite Coulomb fluids, two-point correlation functions exhibit a slow inverse-power law decay along a uniformly charged wall. In this work, we concentrate on the corresponding amplitude function which depends on the distances of the two points from the wall. Recently [L. \v{S}amaj, J. Stat. Phys. {\bf 161}, 227 (2015)], applying a technique of anticommuting variables to a 2D system of charged rectilinear wall with "counter-ions only", we derived a relation between the amplitude function and the density profile which holds for any temperature. In this paper, using the M\"obius conformal transformation of particle coordinates in a disc, a new relation between the amplitude function and the density profile is found for that model. This enables us to prove, at any temperature, the factorization property of the amplitude function in point distances from the wall and to express it in terms of the density profile. Presupposing the factorization property of the amplitude function and using specific sum rules for semi-infinite geometries, a relation between the amplitude function of the charge-charge structure function and the charge profile is derived for many-component Coulomb fluids in any dimension.