Bounded fluctuations and translation symmetry breaking in one-dimensional particle systems

We present general results for one-dimensional systems of point charges (si gned point measures) on the line with a translation invariant distribution mu for which the variance of the total charge in an interval is uniformly b ounded (instead of increasing with the interval length). When the charges a re restricted to multiples of a common unit, and their average charge densi ty does not vanish, then the boundedness of the variance implies translatio n-symmetry breaking - in the sense that there exists a function of the char ge configuration that is nontrivially periodic under translations - and hen ce that mu is not "mixing". Analogous results are formulated also for one d imensional lattice systems under some constraints on the values of the char ges at tile lattice sites and their averages. The general results apply to one-dimensional Coulomb systems, and to certain spin chains, putting on com mon grounds different instances of symmetry breaking encountered there.