Journal of Statistical Physics

, Volume 83, Issue 3–4, pp 761–765 | Cite as

Ill-defined block-spin transformations at arbitrarily high temperatures

  • Aernout C. D. van Enter
Short Communications

Abstract

Examples are presented of block-spin transformations which map the Gibbs measures of the Ising model in two or more dimensions at temperature intervals extending to arbitrarily high temperatures onto non-Gibbsian measures. In this way we provide the first example of this kind of pathology for very high temperatures, and as a corollary also the first example of such a pathology happening at a critical point.

Key Words

Non-Gibbsian measures block-spin map pathology 

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References

  1. 1.
    G. Benfatto, E. Marinari, and E. Olivieri, Some numerical results on the block spin transformation for the 2d Ising model at the critical point,J. Stat. Phys. 78:731–757 (1995).Google Scholar
  2. 2.
    C. Cammarota, The large block spin interaction,Nuovo Cimento B 96:1–16 (1986).Google Scholar
  3. 3.
    E. A. Carlen and A. Soffer, Entropy production by block spin summation and central limit theorems,Commun. Math. Phys. 140:339–371 (1992).Google Scholar
  4. 4.
    M. Cassandro and G. Gallavotti, The Lavoisier law and the critical point,Nuovo Cimento B 25:691–705 (1975).Google Scholar
  5. 5.
    R. L. Dobrushin and S. B. Shlosman, Completely analytical interactions: Constructive description,J. Stat. Phys. 46:983–1014 (1987).Google Scholar
  6. 6.
    R. Fernández and C.-Ed. Pfister, Non-quasilocality of projections of Gibbs measures, EPFL preprint, (1994).Google Scholar
  7. 7.
    R. B. Griffiths and P. A. Pearce, Position-space renormalization-group transformations: Some proofs and some problems,Phys. Rev. Lett. 41:917–920, (1978).Google Scholar
  8. 8.
    R. B. Griffiths and P. A. Pearce, Mathematical properties of position-space renormalization-group transformations,J. Stat. Phys. 20:499–545 (1979).Google Scholar
  9. 9.
    D. Iagolnitzer and B. Souillard, Random fields and limit theorems, inRandom Fields (Esztergom, 1979), J. Fritz, J. L. Lebowitz, and D. Szász, ed. (North-Holland, Amsterdam, 1981), Vol. II, pp. 573–592.Google Scholar
  10. 10.
    R. B. Israel, Banach algebras and Kadanoff transformations, inRandom Fields (Esztergom, 1979), J. Fritz, J. L. Lebowitz, and D. Szász, ed., (North-Holland, Amsterdam, 1981), Vol. II, pp. 593–608.Google Scholar
  11. 11.
    I. A. Kashapov, Justification of the renormalization-group method,Theor. Math. Phys. 42:184–186 (1980).Google Scholar
  12. 12.
    T. Kennedy, Some rigorous results on majority rule renormalization group transformations near the critical point.J. Stat. Phys. 72:15–37 (1993).Google Scholar
  13. 13.
    T. Kennedy, Abstract, Budapest Meeting on Disordered Systems and Statistical physics, Rigorous Results (August 1995), and K. Haller and T. Kennedy, Absence of renormalization group pathologies near the critical temperature—two examples. University of Arizona, preprint.Google Scholar
  14. 14.
    J. Lörinczi, Some results on the projected two-dimensional Ising model, inOn Three Levels, M. Fannes, C. Maes, and A. Verbeure, ed., (Plenum Press, New York, 1994), pp. 373–380.Google Scholar
  15. 15.
    J. Lörinczi, On limits of the Gibbsian formalism in thermodynamics, Ph.D. dissertation, Groningen (1995).Google Scholar
  16. 16.
    J. Lörinczi and K. Vande Velde, A note on the projection of Gibbs measures,J. Stat. Phys. 77:881–887 (1994).Google Scholar
  17. 17.
    C. Maes and K. Vande Velde, The (non-)Gibbsian nature of states invariant under stochastic transformations,Physica A 206:587–603 (1994).Google Scholar
  18. 18.
    F. Martinelli and E. Olivieri, Some remarks on pathologies of renormalization-group transformations,J. Stat. Phys. 72:1169–1177 (1993).Google Scholar
  19. 19.
    F. Martinelli and E. Olivieri, Instability of renormalization-group pathologies under decimation,J. Stat. Phys. 79:25–42 (1995).Google Scholar
  20. 20.
    F. Martinelli and E. Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region I,Commun. Math. Phys. 161:447–486 (1994).Google Scholar
  21. 21.
    F. Martinelli and E. Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region II,Commun. Math. Phys.,161:487–515 (1994).Google Scholar
  22. 22.
    F. Martinelli, E. Olivieri, and R. H. Schonmann, For 2-D lattice spin systems weak mixing implies strong mixing,Commun. Math. Phys. 165:33–48 (1994).Google Scholar
  23. 23.
    C.-Ed. Pfister and K. Vande Velde, Almost sure quasilocality in the random cluster model,J. Stat. Phys. 79:765–774 (1995).Google Scholar
  24. 24.
    J. Salas, Low temperature series for renormalized operators: The ferromagnetic square Ising model,J. Stat. Phys. 80:1309–1326 (1995).Google Scholar
  25. 25.
    R. H. Schonmann and S. B. Shlosman, Complete analyticity for 2D Ising completed,Commun. Math. Phys. 170:453–482 (1995).Google Scholar
  26. 26.
    A. C. D. van Enter, R. Fernández, and R. Kotecký, Pathological behavior of renormalization group maps at high fields and above the transition temperature,J. Stat. Phys. 79:969–992 (1995).Google Scholar
  27. 27.
    A. C. D. van Enter, R. Fernández, and A. D. Sokal, Renormalization transformations in the vicinity of first-order phase transitions: What can and cannot go wrong,Phys. Rev. Lett. 66:3253–3256 (1991).Google Scholar
  28. 28.
    A. C. D. van Enter, R. Fernández, and A. D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory,J. Stat. Phys. 72:879–1167 (1993).Google Scholar
  29. 29.
    K. Vande Velde, Private communication.Google Scholar
  30. 30.
    K. Vande Velde, On the question of quasilocality in large systems of locally interacting components, Ph.D. dissertation, K. U. Leuven (1995).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Aernout C. D. van Enter
    • 1
  1. 1.Institute for Theoretical PhysicsRijksuniversiteit GroningenGroningenThe Netherlands

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