Journal of Statistical Physics

, Volume 39, Issue 1–2, pp 1–13 | Cite as

Absence of phase transitions in certain one-dimensional long-range random systems

  • A. C. D. van Enter
  • J. L. van Hemmen


An Ising chain is considered with a potential of the formJ(i, j)/|ij|α, where theJ(i, j) are independent random variables with mean zero. The chain contains both randomness and frustration, and serves to model a spin glass. A simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature ifα>1. This is to be contrasted with a ferromagnetic interaction which requiresα>2. The basic idea is to prove that the “surface”free energy between two half-lines is finite, although the “surface” energy may be unbounded. Ford-dimensional systems, it is shown that the free energy does not depend on the specific boundary conditions ifα>(1/2)d.

Key words

Phase transition random interactions long-range interactions one-dimensional relative entropy free energy 


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Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • A. C. D. van Enter
    • 1
  • J. L. van Hemmen
    • 1
  1. 1.Universität Heidelberg6900 Heidelberg 1Germany

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