Bounded Fluctuations and Translation Symmetry Breaking: A Solvable Model

The variance of the particle number (equivalently the total charge) in a domain of length L of a one-component plasma (OCP) on a cylinder of circumference W at the reciprocal temperature ;=2, is shown to remain bounded as L . This exactly solvable system with average density \ has a density profile which is periodic with period (\W ) along the axis of the infinitely long cylinder. This illustrates the connection between bounded variance and periodicity in (quasi) one-dimensional systems. When W the system approaches the two-dimensional OCP and the variance in a domain 4 grows like its perimeter | 4|. In this limit, the system is translation invariant with rapid decay of correlations.