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Pathological behavior of renormalization-group maps at high fields and above the transition temeprature

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We show that decimation transformations applied to high-q Potts models result in non-Gibbsian measures even for temperatures higher than the transition temperature. We also show that majority transformations applied to the Ising model in a very strong field at low temperatures produce non-Gibbsian measures. This shows that pathological behavior of renormalization-group transformations is even more widespread than previous examples already suggested.

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  1. 1.

    G. Benfatto, E. Marinari, and E. Olivieri, Some numerical results on the block spin transformation for the2d Ising model at the critical point,J. Stat. Phys. 78:731–757 (1995).

  2. 2.

    S. N. Bernstein,Probability Theory (Gostechizdat, Moscow, 1946).

  3. 3.

    J. Bricmont, K. Kuroda, and J. L. Lebowitz, First order phase transitions in lattice and continuous systems: Extension of Pirogov-Sinai theory.commun. Math. Phys. 101:501–538 (1985).

  4. 4.

    J. T. Chayes, L. Chayes, and J. Fröhlich, The low-temperature behavior of disordered magnets,Commun. Math. Phys. 100:399–437 (1985).

  5. 5.

    L. Chayes, R. Kotecký, and S. B. Shlosman, Aggregation and intermediate phases in dilute spin systems. CTS preprint (1994).

  6. 6.

    R. L. Dobrushin and S. B. Shlosman, Completely analytical interactions: Constructive description.J. Stat. Phys. 46:983–1014 (1987).

  7. 7.

    R. L. Dobrushin and S. B. Shlosman, Private communication (1992).

  8. 8.

    R. Fernández and C.-Ed. Pfister, Non-quasilocality of projections of Gibbs measures, EPFL preprint (1994).

  9. 9.

    C. M. Fortuin, J. Ginibre, and P. W. Kasteleyn, Correlation inequalities on some partially ordered sets,Commun. Math. Phys. 22:89–103 (1971).

  10. 10.

    H.-O. Georgii,Gibbs Measures and Phase Transitions (de Gruyter, Berlin, 1988).

  11. 11.

    R. B. Griffiths, Rigorous results and theorems, InPhase Transitions and Critical Phenomena, Vol. 1 C. Domb and M. S. Green, eds. (Academic Press, New York, 1972).

  12. 12.

    R. B. Griffiths and P. A. Pearce, Position-space renormalization-group transformations: Some proofs and some problems,Phys. Rev. Lett. 41:917–920 (1978).

  13. 13.

    R. B. Griffiths and P. A. Pearce, Mathematical properties of position-space renormalization-group transformations.J. Stat. Phys. 20:499–545 (1979).

  14. 14.

    O. Häggström, Gibbs states and subshifts of finite type, University of Göteborg preprint (1993).

  15. 15.

    O. Häggström, On phase transitions for subshifts of finite type, University of Göteborg preprint (1993).

  16. 16.

    O. Häggström, On the relation between finite range potentials and subshifts of finite type, University of Göteborg preprint (1993).

  17. 17.

    A. Hasenfratz and P. Hazenfratz, Singular renormalization group transformations and first order phase transitions (I),Nucl. Phys. B 295[FS21]:1–20 (1988).

  18. 18.

    N. T. A. Haydn, Classification of Gibbs' states on Smale spaces and one-dimensional lattice systems,Nonlinearity 7:345–366 (1994).

  19. 19.

    I. A. Ibragimov and Yu. V. Linnik,Independent and Stationary Sequences of Random Variables (Wolters-Noordhoff, Groningen, 1971).

  20. 20.

    R. B. Israel, Banach algebras and Kadanoff transformations, J. Fritz, J. L. Lebowitz, and D. Szász, eds.Random Fields (Esztergom, 1979), Vol. II, (North-Holland, Amsterdam, 1981), pp. 593–608.

  21. 21.

    T. Kennedy, Some rigorous results on majority rule renormalization group transformations near the critical point,J. Stat. Phys. 72:15–37 (1993).

  22. 22.

    R. Kotecký and S. B. Shlosman, First-order transitions in large-entropy lattice models,Comm. Math. Phys. 83:493–515 (1982).

  23. 23.

    J. Lörinczi, Some results on the projected two-dimensional Ising model, InOn Three Levels M. Fannes, C. Maes, and A. Verbeure, eds. (Plenum Press, New York, 1994), pp. 373–380.

  24. 24.

    J. Lörinczi and K. Vande Velde, A note on the projection of Gibbs measures.J. Stat. Phys. 77:881–887 (1994).

  25. 25.

    J. Lörinczi and M. Winnink, Some remarks on almost Gibbs states, InCellular Automata and Cooperative Systems, N. Boccara, E. Goles, S. Martinez, and P. Picco, eds. (Kluwer, Dordrecht, 1993), pp. 423–432.

  26. 26.

    F. Martinelli and E. Olivieri, Some remarks on pathologies of renormalization-group transformations,J. Stat. Phys. 72:1169–1177 (1993).

  27. 27.

    F. Martinelli and E. Olivieri, Instability of renormalization-group pathologies under decimation, University of Rome II preprint (1994);J. Stat. Phys. 79:25–42 (1995).

  28. 28.

    J. Salas, Low-temperature series for renormalized operators: The ferromagnetic squarelattice Ising model, New York University preprint mp-arc 94-290 (1994).

  29. 29.

    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).

  30. 30.

    A. C. D. van Enter, R. Fernández, and A. D. Sokal, Regularity properties and patholgies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory,J. Stat. Phys. 72:879–1167 (1993).

  31. 31.

    K. Vande Velde. Private communication.

  32. 32.

    M. Zahradník, On the structure of low temperature phases in three-dimensional spin models with random impurities: A general Pirogov-Sinai approach, InPhase Transitions: Mathematics, Physics, Biology,..., R. Kotecký, ed. (World Scientific, Singapore, 1993), pp. 225–237.

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van Enter, A.C.D., Fernández, R. & Kotecký, R. Pathological behavior of renormalization-group maps at high fields and above the transition temeprature. J Stat Phys 79, 969–992 (1995). https://doi.org/10.1007/BF02181211

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Key Words

  • Non-Gibbsian measures
  • real-space renormalization
  • complete analyticity
  • majority-rule
  • decimation