Advertisement

Communications in Mathematical Physics

, Volume 46, Issue 2, pp 167–182 | Cite as

The GHS and other correlation inequalities for a class of even ferromagnets

  • Richard S. Ellis
  • James L. Monroe
  • Charles M. Newman
Article

Abstract

We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin −1/2 Ising models, ϕ4 field theories, and other continuous spin models. The proofs are based on the properties of a classG of probability measures which contains all measures of the form const exp(−V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ∞). A new proof of the GKS inequalities using similar ideas is also given.

Keywords

Neural Network Statistical Physic Field Theory Complex System Quantum Field Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blume, M., Emery, V. J., Griffiths, R. B.: Ising model for the λ transition and phase separation in He3-He4 mixtures. Phys. Rev. A4, 1071–1077 (1971)Google Scholar
  2. 2.
    Ellis, R. S.: Concavity of magnetization for a class of even ferromagnets. Bull A.M.S.81, 925–929 (1975)Google Scholar
  3. 3.
    Ellis, R. S., Monroe, J. L.: A simple proof of the GHS and further inequalities. Commun. math. Phys.41, 33–38 (1975)Google Scholar
  4. 4.
    Ginibre, J.: General formulation of Griffiths inequalities. Commun. math. Phys.16, 310–328 (1970)Google Scholar
  5. 5.
    Glimm, J., Jaffe, A.: Absolute bounds on vertices and couplings. Rockefeller Univ. and Harvard Univ. (preprint) (1974)Google Scholar
  6. 6.
    Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(ϕ)2 model and other applications of high temperature expansions. In: Velo, G., Wightman, A. S. (Eds.): Constructive quantum field theory, p. 133–198. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  7. 7.
    Glimm, J., Jaffe, A., Spencer, T.: The Wightman axioms and particle structure in theP(ϕ)2 quantum field model. Ann. Math.100, 585–632 (1974)Google Scholar
  8. 8.
    Griffiths, R. B.: Correlation in Ising ferromagnets. J. Math. Phys.8, 478–483 (1967)Google Scholar
  9. 9.
    Griffiths, R. B.: Rigorous results for Ising ferromagnets of arbitrary spins. J. Math. Phys.10, 1559–1565 (1969)Google Scholar
  10. 10.
    Griffiths, R. B.: Thermodynamics near the two-fluid critical mixing point in He3-He4. Phys. Rev. Letters24, 715–717 (1970)Google Scholar
  11. 11.
    Griffiths, R. B., Hurst, C. A., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys.11, 790–795 (1970)Google Scholar
  12. 12.
    Griffiths, R. B., Simon, B.: The (ϕ4)2 field theory as a classical Ising model. Commun. math. Phys.33, 145–164 (1973)Google Scholar
  13. 13.
    Kelley, D., Sherman, S.: General Griffiths inequalities on correlations in Ising ferromagnets. J. Math. Phys.9, 466–484 (1968)Google Scholar
  14. 14.
    Lebowitz, J. L.: Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems. Commun. math. Phys.28, 313–321 (1972)Google Scholar
  15. 15.
    Lebowitz, J. L.: GHS and other inequalities. Commun. math. Phys.35, 87–92 (1974)Google Scholar
  16. 16.
    Mukamel, D., Blume, M.: Ising models for tricritical points in ternary mixtures. Phys. Rev. A10, 610–617 (1974)Google Scholar
  17. 17.
    Monroe, J. L., Siegert, A. J. F.: GKS inequalities for arbitrary spin Ising ferromagnets. J. Stat. Phys.10, 237–244 (1974)Google Scholar
  18. 18.
    Newman, C. M.: Gaussian correlation inequalities for ferromagnets. Z. f. Wahrscheinlichkeitsth. (to appear)Google Scholar
  19. 19.
    Newman, C. M.: Inequalities for Ising models and field theories which obey the Lee-Yang theorem. Commun. math. Phys.41, 1–9 (1975)Google Scholar
  20. 20.
    Newman, C. M.: Zeroes of the partition function for generalized Ising systems. Comm. Pure Appl. Math.27, 143–159 (1974)Google Scholar
  21. 21.
    Newman, C. M.: Moment inequalities for ferromagnetic Gibbs distributions. J. Math. Phys. (to appear)Google Scholar
  22. 22.
    Percus, J.: Correlation inequalities for Ising spin lattices. Commun. math. Phys.40, 283–308 (1975)Google Scholar
  23. 23.
    Preston, C.: An application of the GHS inequalities to show the absence of phase transitions for Ising spin systems. Commun. math. Phys.35, 253–255 (1974)Google Scholar
  24. 24.
    Simon, B.: Approximation of Feynman integrals and Markov fields by spin systems. Proc. of Internatl. Congr. of Mathematicians (Vancouver, B.C., 1974)Google Scholar
  25. 25.
    Simon, B.: Bose quantum field theory as an Ising ferromagnet: recent developments. Princeton Univ. (preprint) (1975)Google Scholar
  26. 26.
    Simon, B.: TheP(ϕ)2 Euclidean (quantum) field theory. Princeton, N.J.: Princeton University Press 1974Google Scholar
  27. 27.
    Sylvester, G.: Continuous-spin inequalities for Ising ferromagnets. M.I.T. (preprint) (1975)Google Scholar
  28. 28.
    Sylvester, G.: Private communicationGoogle Scholar
  29. 29.
    Sylvester, G.: Representations and inequalities for Ising model Ursell functions. M.I.T. (preprint) (1974)Google Scholar
  30. 30.
    Thompson, C.: Mathematical statistical mechanics. New York: Macmillan 1972Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Richard S. Ellis
    • 1
  • James L. Monroe
    • 2
  • Charles M. Newman
    • 3
  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA
  2. 2.Department of PhysicsNorthwestern UniversityEvanstonUSA
  3. 3.Department of MathematicsIndiana UniversityBloomingtonUSA

Personalised recommendations