Charge Correlations in a Coulomb System Along a Plane Wall: a Relation Between Asymptotic Behavior and Dipole Moment

Classical Coulomb systems at equilibrium, bounded by a plane dielectric wall, are studied. A general two-point charge correlation function is considered. Valid for any fixed position of one of the points, a new relation is found between the algebraic tail of the correlation function along the wall and the dipole moment of that function. The relation is tested first in the weak-coupling (Debye-H\"uckel) limit, and afterwards, for the special case of a plain hard wall, on the exactly solvable two-dimensional two-component plasma at coupling $\Gamma=2$, and on the two-dimensional one-component plasma at an arbitrary even integer $\Gamma$.