 00293 M. Aizenman, S. Goldstein and J.L. Lebowitz
 Bounded Fluctuations and Translation Symmetry Breaking in OneDimensional Particle Systems
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Jul 14, 00

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Abstract. We present general results for onedimensional systems of point charges (signed point measures) on the line with a translation invariant distribution $\mu$ for which the variance of the total charge in an interval is uniformly bounded (instead of increasing with the interval length). When the charges are restricted to multiples of a common unit, and their average charge density does not vanish, then the boundedness of the variance implies translationsymmetry breaking  in the sense that there exists a function of the charge configuration that is nontrivially periodic under translations  and hence that $\mu$ is not ``mixing.'' Analogous results are formulated also for one dimensional lattice systems under some constraints on the values of the charges at the lattice sites and their averages. The general results apply to onedimensional Coulomb systems, and to certain spin chains, putting on common grounds different instances of symmetry breaking encountered there.
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