Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Our main results apply to local (in position space) RG maps acting on systems of bounded spins (compact single-spin space). Regarding regularity, we show that the RG map, defined on a suitable space of interactions (=formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce, and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension d⩾3, these pathologies occur in a full neighborhood {β>β 0, ¦h¦<ε(β)} of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension d⩾2, the pathologies occur at low temperatures for arbitrary magnetic field strength. Pathologies may also occur in the critical region for Ising models in dimension d⩾4. We discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems. In addition, we discuss critically the concept of Gibbs measure, which is at the heart of present-day classical statistical mechanics. We provide a careful, and, we hope, pedagogical, overview of the theory of Gibbsian measures as well as (the less familiar) non-Gibbsian measures, emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations. We give a rather complete catalogue of the known examples of such occurrences. The main message of this paper is that, despite a well-established tradition, Gibbsiannessshould not be taken for granted.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Aizenman, Translation invariance and instability of phase coexistence in the two dimensional Ising system,Commun. Math. Phys. 73:83–94 (1980).

    Google Scholar 

  2. M. Aizenman, Geometric analysis ofϕ 44 fields and Ising models. Parts I and II,Commun. Math. Phys. 86:1–48 (1982).

    Google Scholar 

  3. M. Aizenman, The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory,Commun. Math. Phys. 97:91–110 (1985).

    Google Scholar 

  4. M. Aizenman, J. Bricmont, and J. L. Lebowitz, Percolation of the minority spins in high-dimensional Ising models,J. Stat. Phys. 49:859–865 (1987).

    Google Scholar 

  5. M. Aizenman, J. T. Chayes, L. Chayes, J. Fröhlich, and L. Russo, On a sharp transition from area law to perimeter law in a system of random surfaces,Commun. Math. Phys. 92:19–69 (1983).

    Google Scholar 

  6. M. Aizenman, J. T. Chayes, L. Chayes, and C. M. Newman, Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models,J. Stat. Phys. 50:1–40 (1988).

    Google Scholar 

  7. M. Aizenman and R. Fernández, On the critical behavior of the magnetization in high-dimensional Ising models,J. Stat. Phys. 44:393–454 (1986).

    Google Scholar 

  8. M. Aizenman and R. Graham, On the renormalized coupling constant and the susceptibility inϕ 44 field theory and the Ising model in four dimensions,Nucl. Phys. B 225[FS9]:261–288 (1983).

    Google Scholar 

  9. M. Aizenman and E. H. Lieb, The third law of thermodynamics and the degeneracy of the ground state for lattice systems,J. Stat. Phys. 24:279–297 (1981).

    Google Scholar 

  10. M. P. Almeida and B. Gidas, A variational method for estimating the parameters of MRF from complete or incomplete data,Ann. Appl. Probab. 3:103–136 (1993).

    Google Scholar 

  11. D. J. Amit and L. Peliti, On dangerous irrelevant operators,Ann. Phys. 140:207–231 (1982).

    Google Scholar 

  12. P. W. Anderson and G. Yuval, Some numerical results on the Kondo problem and the inverse-square one-dimensional Ising model,J. Phys. C 4:607–620 (1971).

    Google Scholar 

  13. P. W. Anderson, G. Yuval, and D. R. Hamann, Exact results in the Kondo problem. II. Scaling theory, qualitatively correct solution, and some new results on one-dimensional classical statistical models,Phys. Rev. B 1:4464–4473 (1970).

    Google Scholar 

  14. C. Aragão de Carvalho, S. Caracciolo, and J. Fröhlich, Polymers and gϕ4 theory in four dimensions,Nucl. Phys. B 215[FS7]:209–248 (1983).

    Google Scholar 

  15. M. Asorey, J. G. Esteve, R. Fernández, and J. Salas, Rigorous analysis of renormalization group pathologies in the 4-state clock model,Nucl. Phys. B 392:593–618 (1993).

    Google Scholar 

  16. M. B. Averintsev, Gibbs description of random fields whose conditional probabilities may vanish,Probl. Inform. Transmission 11:326–334 (1975).

    Google Scholar 

  17. R. R. Bahadur,Some Limit Theorems in Statistics (SIAM, Philadelphia, Pennsylvania, 1971).

    Google Scholar 

  18. G. A. Baker and S. Krinsky, Renormalization group structure of translation invariant ferromagnets,J. Math. Phys. 18:590–607 (1977).

    Google Scholar 

  19. T. Balaban, Ultraviolet stability in field theory. Theφ 43 model, inScaling and Self-Similarity in Physics, J. Fröhlich, ed. (Birkhäuser, Basel, 1983).

    Google Scholar 

  20. T. Balaban, Large field renormalization. II. Localization, exponentiation and bounds for theR operation,Commun. Math. Phys. 122:355–392 (1989).

    Google Scholar 

  21. R. Balian,From Microphysics to Macrophysics: Methods and Applications of Statistical Physics (Springer-Verlag, Berlin, 1991).

    Google Scholar 

  22. A. G. Basuev, Complete phase diagrams with respect to external fields at low temperatures for models with nearest-neighbor interaction in the case of a finite or countable number of ground states,Theor. Math. Phys. 58:171–182 (1984).

    Google Scholar 

  23. A. G. Basuev, Hamiltonian of the phase separation border and phase transitions of the first kind. I,Theor. Math. Phys. 64:716–734 (1985).

    Google Scholar 

  24. A. G. Basuev, Hamiltonian of the phase separation border and phase transitions of the first kind. II. The simplest disordered phases,Theor. Math. Phys. 72:861–871 (1987).

    Google Scholar 

  25. G. A. Battle and L. Rosen, The FKG inequality for the Yukawa2 quantum field theory,J. Stat. Phys. 22:123–192 (1980).

    Google Scholar 

  26. H. Bauer,Probability Theory and Elements of Measure Theory (Holt, Rinehart and Winston, New York, 1972).

    Google Scholar 

  27. R. T. Baumel, On spontaneously broken symmetry in the P(φ)2 model quantum field theory, Ph.D. thesis, Princeton University (June 1979).

  28. T. L. Bell and K. G. Wilson, Finite-lattice approximations to renormalization groups,Phys. Rev. B 11:3431–3444 (1975).

    Google Scholar 

  29. G. Benfatto, M. Cassandro, G. Gallavotti, F. Nicolò, E. Olivieri, E. Presutti, and E. Scacciatelli, Ultraviolet stability in Euclidean scalar field theories,Commun. Math. Phys. 71:95–130 (1980).

    Google Scholar 

  30. P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).

    Google Scholar 

  31. P. Billingsley,Probability and Measure (Wiley, New York, 1979).

    Google Scholar 

  32. M. J. Bissett, A perturbation-theoretic derivation of Wilson's ‘incomplete integration’ theory,J. Phys. C 6:3061–3070 (1973).

    Google Scholar 

  33. P. M. Bleher and P. Major, Critical phenomena and universal exponents in statistical physics. On Dyson's hierarchical model,Ann. Prob. 15:431–477 (1987).

    Google Scholar 

  34. H. W. J. Blöte and R. H. Swendsen, First-order phase transitions and the three-state Potts model,Phys. Rev. Lett. 43:799–802 (1979).

    Google Scholar 

  35. E. Bolthausen, Markov process large deviations in the τ-topology,Stoch. Proc. Appl. 25:95–108 (1987).

    Google Scholar 

  36. C. Borgs and J. Z. Imbrie, A unified approach to phase diagrams in field theory and statistical mechanics,Commun. Math. Phys. 123:305–328 (1989).

    Google Scholar 

  37. C. Borgs and R. Kotecký, A rigorous theory of finite-size scaling at first-order phase transitions,J. Stat. Phys. 61:79–119 (1990).

    Google Scholar 

  38. H. J. Brascamp and E. H. Lieb, Some inequalities for Gaussian measures, inFunctional Integration and its Applications, A. M. Arthurs, ed. (Clarendon Press, Oxford, 1975).

    Google Scholar 

  39. H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation,J. Funct. Anal. 22:366–389 (1976).

    Google Scholar 

  40. H. J. Brascamp, E. H. Lieb, and J. L. Lebowitz, The statistical mechanics of anharmonic lattices,Bull. Int. Statist. Inst. 46 (Book 1):393–404 (1975).

    Google Scholar 

  41. O. Bratteli and D. W. Robinson,Operator Algebras and Quantum Statistical Mechanics II (Springer-Verlag, Berlin, 1981).

    Google Scholar 

  42. J. Bricmont, J.-R. Fontaine, and L. J. Landau, On the uniqueness of the equilibrium state for plane rotators,Commun. Math. Phys. 56:281–290 (1977).

    Google Scholar 

  43. J. Bricmont, J.-R. Fontaine, J. L. Lebowitz, and T. Spencer, Lattice systems with a continuous symmetry I. Perturbation theory for unbounded spins,Commun. Math. Phys. 78:281–302 (1980).

    Google Scholar 

  44. J. Bricmont and A. Kupiainen, Lower critical dimension for the random field Ising model,Phys. Rev. Lett. 59:1829–1832 (1987).

    Google Scholar 

  45. J. Bricmont and A. Kupiainen, Phase transition in the 3d random field Ising model,Commun. Math. Phys. 116:539–572 (1988).

    Google Scholar 

  46. J. Bricmont, K. Kuroda, and J. L. Lebowitz, The structure of Gibbs states and phase coexistence for non-symmetric continuum Widom-Rowlinson models,Z. Wahrsch. verw. Geb. 67:121–138 (1984).

    Google Scholar 

  47. 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).

    Google Scholar 

  48. J. Bricmont and J. Slawny, First order phase transitions and perturbation theory, inStatistical Mechanics and Field Theory: Mathematical Aspects (Proceedings, Groningen 1985, Lecture Notes in Physics #257, T. C. Dorlas, N. M. Hugenholtz, and M. Winnink, eds. (Springer-Verlag, Berlin, 1986).

    Google Scholar 

  49. J. Bricmont and J. Slawny, Phase transitions in systems with a finite number of dominant ground states,J. Stat. Phys. 54:89–161 (1989).

    Google Scholar 

  50. D. A. Browne and P. Kleban, Equilibrium statistical mechanics for kinetic phase transitions,Phys. Rev. A 40:1615–1626 (1989).

    Google Scholar 

  51. A. D. Bruce and A. Aharony, Coupled order parameters, symmetry-breaking irrelevant scaling fields, and tetracritical points,Phys. Rev. B 11:478–499 (1975).

    Google Scholar 

  52. W. Bryc, On the large deviation principle for stationary weakly dependent random fields,Ann. Prob. 20:1004–1030 (1992).

    Google Scholar 

  53. D. Brydges and T. Spencer, Self-avoiding walks in 5 or more dimensions,Commun. Math. Phys. 97:125–148 (1985).

    Google Scholar 

  54. D. Brydges and H.-T. Yau, Gradϕ perturbations of massless Gaussian fields,Commun. Math. Phys. 129:351–392 (1990).

    Google Scholar 

  55. T. W. Burkhardt, Random-field singularities in position-space renormalization-group transformations,Phys. Rev. Lett. 43:1629–1631 (1979).

    Google Scholar 

  56. T. W. Burkhardt and J. M. J. van Leeuwen, Progress and problems in real-space renormalization, inReal-Space Renormalization, T. W. Burkhardt and J. M. J. van Leeuwen, eds. (Springer-Verlag, Berlin, 1982).

    Google Scholar 

  57. C. Cammarota, The large block spin interaction,Nuovo Cim. B 96:1–16 (1986).

    Google Scholar 

  58. J. L. Cardy, One-dimensional models with 1/r 2 interactions,J. Phys. A 14:1407–1415 (1981).

    Google Scholar 

  59. 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 

  60. M. Cassandro and E. Olivieri, Renormalization group and analyticity in one dimension: A proof of Dobrushin's theorem,Commun. Math. Phys. 80:255–269 (1981).

    Google Scholar 

  61. M. Cassandro, E. Olivieri, A. Pellegrinotti, and E. Presutti, Existence and uniqueness of DLR measures for unbounded spin systems,Z. Wahrsch. verw. Geb. 41:313–334 (1978).

    Google Scholar 

  62. N. N. Čencov,Statistical Decision Rules and Optimal Inference (American Mathematical Society, Providence, Rhode Island, 1982).

    Google Scholar 

  63. S. Chowla and M. Cowles, Remarks on equations related to Fermat's last theorem, inNumber Theory Related to Fermat's Last Theorem, N. Koblitz, ed. (Birkhäuser, Basel, 1982).

    Google Scholar 

  64. D. V. Chudnovsky and G. V. Chudnovsky, Transcendental methods and theta-functions, inProceedings of Symposia in Pure Mathematics, Vol. 49, Part 2 (American Mathematical Society, Providence, Rhode Island, 1989), pp. 167–232.

    Google Scholar 

  65. F. S. Cohen and D. B. Cooper, Simple parallel hierarchical and relaxation algorithms for segmenting noncausal Markovian random fields,IEEE Trans. Pattern Anal. Machine Intell. 9:195–219 (1987).

    Google Scholar 

  66. J. E. Cohen, Y. Iwasa, G. Rautu, M. B. Ruskai, E. Seneta, and G. Zbaganu, Relative entropy under mappings by stochastic matrices,Lin. Alg. Appl. 179:211–235 (1993).

    Google Scholar 

  67. F. Comets, Grandes déviations pour des champs de Gibbs sur ℤd,C. R. Acad. Sci. Paris I 303:511–513 (1986).

    Google Scholar 

  68. I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai,Ergodic Theory (Springer-Verlag, Berlin, 1982).

    Google Scholar 

  69. I. Csiszár, Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten,Magyar Tud. Akad. Mat. Kutató Int. Közl. 8:85–108 (1963) [see alsoMath. Rev. 29, #1671 (1965)].

    Google Scholar 

  70. I. Csiszár,I-divergence geometry of probability distributions and minimization problems,Ann. Prob. 3:146–158 (1975).

    Google Scholar 

  71. I. Csiszár, Sanov property, generalized I-projection and a conditional limit theorem,Ann. Prob. 12:768–793 (1984).

    Google Scholar 

  72. H. A. M. Daniëls and A. C. D. van Enter, Differentiability properties of the pressure in lattice systems,Commun. Math. Phys. 71:65–76 (1980).

    Google Scholar 

  73. J. de Coninck and C. M. Newman, The magnetization-energy scaling limit in high dimension,J. Stat. Phys. 59:1451–1467 (1990).

    Google Scholar 

  74. K. Decker, A. Hasenfratz, and P. Hasenfratz, Singular renormalization group transformations and first order phase transitions (II). Monte Carlo renormalization group results,Nucl. Phys. B 295[FS21]:21–35 (1988).

    Google Scholar 

  75. C. Dellacherie and P.-A. Meyer,Probabilities and Potential (North-Holland, Amsterdam, 1978).

    Google Scholar 

  76. P. Dénes, Über die Diophantische Gleichungx l +y l =cz l,Acta Math. 88:241–251 (1952).

    Google Scholar 

  77. J.-D. Deuschel and D. W. Stroock,Large Deviations (Academic Press, San Diego, 1989).

    Google Scholar 

  78. L. E. Dickson,History of the Theory of Numbers, Vol. II (Chelsea, New York, 1971).

  79. E. L. Dinaburg and A. E. Mazel, Low-temperature phase transitions in ANNNI model, inProceedings 8th International Congress on Mathematical Physics, M. Mebkhout and R. Sénéor, eds. (World Scientific, Singapore, 1987).

    Google Scholar 

  80. E. L. Dinaburg and A. E. Mazel, Analysis of low-temperature phase diagram of the microemulsion model,Commun. Math. Phys. 125:25–42 (1989).

    Google Scholar 

  81. E. L. Dinaburg, A. E. Mazel, and Ya. G. Sinai, ANNNI model and contour models with interactions, inMathematical Physics Reviews, Soviet Science Reviews Section C, Vol. 6, S. P. Novikov, ed. (Gordon and Breach, New York, 1986).

    Google Scholar 

  82. E. L. Dinaburg and Ya. G. Sinai, An analysis of ANNNI model by Peierl's [sic] contour method,Commun. Math. Phys. 98:119–144 (1985).

    Google Scholar 

  83. P. G. L. Dirichlet, Mémoire sur l'impossibilité de quelques équations indéterminées du cinquième degré,J. Reine Angew. Math. (Crelle's J.) 3:354–376 (1828).

    Google Scholar 

  84. R. L. Dobrushin, Existence of a phase transition in the two-dimensional and three-dimensional Ising models,Sov. Phys. Doklady 10:111–113 (1965).

    Google Scholar 

  85. R. L. Dobrushin, The description of a random field by means of conditional probabilities and conditions of its regularity,Theor. Prob. Appl. 13:197–224 (1968).

    Google Scholar 

  86. R. L. Dobrushin, The problem of uniqueness of a Gibbs random field and the problem of phase transitions,Funct. Anal. Appl. 2:302–312 (1968).

    Google Scholar 

  87. R. L. Dobrushin, Gibbs states describing coexistence of phases for a three-dimensional Ising model,Theor. Prob. Appl. 17:582–600 (1972).

    Google Scholar 

  88. R. L. Dobrushin, Gaussian random fields—Gibbsian point of view, inMulticomponent Random Systems, R. L. Dobrushin and Ya. G. Sinai, eds. (Marcel Dekker, New York, 1980), pp. 119–152.

    Google Scholar 

  89. R. L. Dobrushin, J. Kolafa, and S. B. Shlosman, Phase diagram of the two-dimensional Ising antiferromagnet (computer-assisted proof),Commun. Math. Phys. 102:89–103 (1985).

    Google Scholar 

  90. R. L. Dobrushin and M. R. Martirosyan, Nonfinite perturbations of Gibbs fields,Theor. Math. Phys. 74:10–20 (1988).

    Google Scholar 

  91. R. L. Dobrushin and M. R. Martirosyan, Possibility of high-temperature phase transitions due to the many-particle nature of the potential,Theor. Math. Phys. 75:443–448 (1988).

    Google Scholar 

  92. R. L. Dobrushin and E. A. Pecherski, Uniqueness condition for finitely dependent random fields, inRandom Fields. Esztergom (Hungary) 1979, Vol. I (North-Holland, Amsterdam, 1981).

    Google Scholar 

  93. R. L. Dobrushin and E. A. Pecherski, A criterion for the uniqueness of Gibbsian fields in the non-compact case, inProbability Theory and Mathematical Statistics, Lecture Notes in Mathematics #1021, (Springer-Verlag, Berlin, 1983), pp. 97–110.

    Google Scholar 

  94. R. L. Dobrushin and S. B. Shlosman, Nonexistence of one- and two-dimensional Gibbs fields with noncompact group of continuous symmetries, inMulticomponent Random Systems, R. L. Dobrushin and Ya. G. Sinai, eds. (Marcel Dekker, New York, 1980), pp. 199–210.

    Google Scholar 

  95. R. L. Dobrushin and S. B. Shlosman, Completely analytic random fields, inStatistical Mechanics and Dynamical Systems, J. Fritz, A. Jaffe and D. Szàsz, eds. (Birkhäuser, Basel, 1985).

    Google Scholar 

  96. R. L. Dobrushin and S. B. Shlosman, Constructive criterion for the uniqueness of a Gibbs field, inStatistical Mechanics and Dynamical Systems, J. Fritz, A. Jaffe and D. Szàsz, eds. (Birkhäuser, Basel, 1985).

    Google Scholar 

  97. R. L. Dobrushin and S. B. Shlosman, The problem of translation invariance of Gibbs states at low temperatures, inMathematical Physics Reviews, Soviet Science Reviews Section C, Vol. 5, S. P. Novikov, ed. (Gordon and Breach, New York, 1985).

    Google Scholar 

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

    Google Scholar 

  99. R. L. Dobrushin and M. Zahradnik, Phase diagrams for continuous spin systems, inMathematical Problems of Statistical Physics and Dynamics, R. L. Dobrushin, ed. (Reidel, Dordrecht, 1985).

    Google Scholar 

  100. Y. Domar, On the Diophantine equation ¦Ax n −By n¦=1,Math. Scand. 2:29–32 (1954).

    Google Scholar 

  101. M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, I,Commun. Pure Appl. Math. 28:1–47 (1975).

    Google Scholar 

  102. M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, II,Commun. Pure Appl. Math. 28:279–301 (1975).

    Google Scholar 

  103. M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, III,Commun. Pure Appl. Math. 28:389–461 (1976).

    Google Scholar 

  104. M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, IV,Commun. Pure Appl. Math. 36:183–212 (1983).

    Google Scholar 

  105. T. C. Dorlas and A. C. D. van Enter, Non-Gibbsian limit for large-block majority-spin transformations,J. Stat. Phys. 55:171–181 (1989).

    Google Scholar 

  106. N. Dunford and J. T. Schwartz,Linear Operators (Interscience, New York, 1958).

    Google Scholar 

  107. F. Dunlop, Correlation inequalities for multicomponent rotators,Commun. Math. Phys. 49:247–256 (1976).

    Google Scholar 

  108. F. Dunlop and C. M. Newman, Multicomponent field theories and classical rotators,Commun. Math. Phys. 44:223–235 (1975).

    Google Scholar 

  109. E. B. Dynkin, Sufficient statistics and extreme points,Ann. Prob. 6:705–730 (1978).

    Google Scholar 

  110. R. E. Edwards,Fourier Series: A Modern Introduction, Vol. I (Holt, Rinehart and Winston, New York, 1967).

    Google Scholar 

  111. R. G. Edwards and A. D. Sokal, Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm,Phys. Rev. D 38:2009–2012 (1988).

    Google Scholar 

  112. R. S. Ellis,Entropy, Large Deviations and Statistical Mechanics (Springer-Verlag, Berlin, 1985).

    Google Scholar 

  113. J. H. Evertse,Upper Bounds for the Numbers of Solutions of Diophantine Equations (Mathematisch Centrum, Amsterdam, 1983).

    Google Scholar 

  114. G. Felder and J. Fröhlich, Intersection properties of simple random walks: A renormalization group approach,Commun. Math. Phys. 97:111–124 (1985).

    Google Scholar 

  115. B. U. Felderhof, Phase transitions in one-dimensional cluster-interaction fluids. III. Correlation functions,Ann. Phys. 58:281–300 (1970).

    Google Scholar 

  116. B. U. Felderhof and M. E. Fisher, Phase transitions in one-dimensional cluster-interaction fluids. II. Simple logarithmic model,Ann. Phys. 58:268–280 (1970).

    Google Scholar 

  117. R. Fernández, J. Fröhlich, and A. D. Sokal,Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory (Springer-Verlag, Berlin, 1992).

    Google Scholar 

  118. M. E. Fisher, On discontinuity of the pressure,Commun. Math. Phys. 26:6–14 (1972).

    Google Scholar 

  119. M. E. Fisher, Crossover effects and operator expansions, inRenormalization Group in Critical Phenomena and Quantum Field Theory: Proceedings of a Conference, J. D. Gunton and M. S. Green, eds. (Temple University, Philadelphia, Pennsylvania, 1974), pp. 65–68.

    Google Scholar 

  120. M. E. Fisher, Scaling, universality and renormalization group theory, inCritical Phenomena (Stellenbosch 1982), Lecture Notes in Physics # 186, F. J. W. Hahne, ed. (Springer-Verlag, Berlin, 1983), pp. 1–139.

    Google Scholar 

  121. M. E. Fisher and A. N. Berker, Scaling for first-order phase transitions in thermodynamic and finite systems,Phys. Rev. B 26:2507–2513 (1982).

    Google Scholar 

  122. M. E. Fisher and B. U. Felderhof, Phase transitions in one-dimensional cluster-interaction fluids. IA. Thermodynamics,Ann. Phys. 58:176–216 (1970).

    Google Scholar 

  123. M. E. Fisher and B. U. Felderhof, Phase transitions in one-dimensional cluster-interaction fluids. IB. Critical behavior,Ann. Phys. 58:217–267 (1970).

    Google Scholar 

  124. M. E. Fisher and S. Sarbach, Nonuniversality of tricritical behavior,Phys. Rev. Lett. 41:1127–1130 (1978).

    Google Scholar 

  125. H. Föllmer, On entropy and information gain in random fields,Z. Wahrsch. verw. Geb. 26:207–217 (1973).

    Google Scholar 

  126. H. Föllmer, Random fields and diffusion processes, inÉcole d'été de Probabilités de Saint-Flour XV-XVII, Lecture Notes in Mathematics #1362, P. Hennequin, ed. (Springer-Verlag, Berlin, 1988).

    Google Scholar 

  127. H. Föllmer and S. Orey, Large deviations for the empirical field of a Gibbs measure,Ann. Prob. 16:961–977 (1988).

    Google Scholar 

  128. C. M. Fortuin, On the random cluster model. III. The simple random cluster model,Physica 59:545–570 (1972).

    Google Scholar 

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

    Google Scholar 

  130. C. M. Fortuin and P. W. Kasteleyn, On the random cluster model. I. Introduction and relation to other models,Physica 57:536–564 (1972).

    Google Scholar 

  131. Z. Friedman and J. Felsteiner, Kadanoff block transformation by the Monte-Carlo technique,Phys. Rev. B 15:5317–5319 (1977).

    Google Scholar 

  132. A. Frigessi and M. Piccioni, Parameter estimation for two-dimensional Ising fields corrupted by noise,Stoch. Proc. Appl. 34:297–311 (1990).

    Google Scholar 

  133. J. Fröhlich, On the triviality ofλϕ 44 theories and the approach to the critical point in d ≧ 4 dimensions,Nucl. Phys. B 200[FS4]:281–296 (1982).

    Google Scholar 

  134. J. Fröhlich, Mathematical aspects of the physics of disordered systems, inCritical Phenomena, Random Systems, Gauge Theories [Les Houches 1984], K. Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1986), Part II, pp. 725–893.

    Google Scholar 

  135. J. Fröhlich, R. B. Israel, E. H. Lieb, and B. Simon, Phase transitions and reflection positivity. II. Lattice systems with short-range and Coulomb interactions,J. Stat. Phys. 22:297–347 (1980).

    Google Scholar 

  136. J. Fröhlich and C. E. Pfister, On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems,Commun. Math. Phys. 81:277–298 (1981).

    Google Scholar 

  137. J. Fröhlich and T. Spencer, The Kosterlitz-Thouless phase transition in the two-dimensional plane rotator and Coulomb gas,Phys. Rev. Lett. 46:1006–1009 (1981).

    Google Scholar 

  138. J. Fröhlich and T. Spencer, The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas,Commun. Math. Phys. 81:527–602 (1981).

    Google Scholar 

  139. J. Fröhlich and T. Spencer, Phase diagrams and critical properties of (classical) Coulomb systems, inRigorous Atomic and Molecular Physics, G. Velo and A. S. Wightman, eds. (Plenum Press, New York, 1981), pp. 327–370.

    Google Scholar 

  140. J. Fröhlich and T. Spencer, The phase transition in the one-dimensional Ising model with 1/r 2 interaction energy,Commun. Math. Phys. 84:87–101 (1982).

    Google Scholar 

  141. J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy,Commun. Math. Phys. 88:151–184 (1983).

    Google Scholar 

  142. G. Gallavotti and S. Miracle-Sole, Statistical mechanics of lattice systems,Commun. Math. Phys. 5:317–324 (1967).

    Google Scholar 

  143. G. Gallavotti and S. Miracle-Sole, Correlation functions of lattice systems,Commun. Math. Phys. 7:274–288 (1968).

    Google Scholar 

  144. G. Gallavotti and S. Miracle-Sole, Equilibrium states of the Ising model in the two-phase region,Phys. Rev. B 5:2555–2559 (1972).

    Google Scholar 

  145. J. M. Gandhi, On Fermat's last theorem,Am. Math. Monthly 71:998–1006 (1964).

    Google Scholar 

  146. P. Gänssler, Compactness and sequential compactness in spaces of measures,Z. Wahrsch. verw. Geb. 17:124–146 (1971).

    Google Scholar 

  147. P. L. Garrido, A. Labarta, and J. Marro, Stationary nonequilibrium states in the Ising model with locally competing temperatures,J. Stat. Phys. 49:551–568 (1987).

    Google Scholar 

  148. K. Gawedzki, Rigorous renormalization group at work,Physica 140A:78–84 (1986).

    Google Scholar 

  149. K. Gawedzki, R. Kotecký, and A. Kupiainen, Coarse graining approach to first order phase transitions,J. Stat. Phys. 47:701–724 (1987).

    Google Scholar 

  150. K. Gawedzki and A. Kupiainen, A rigorous block spin approach to massless lattice theories,Commun. Math. Phys. 77:31–64 (1980).

    Google Scholar 

  151. K. Gawedzki and A. Kupiainen, Block spin renormalization group for dipole gas andΔφ)4,Ann. Phys. 147:198–243 (1983).

    Google Scholar 

  152. K. Gawedzki and A. Kupiainen, Gross-Neveu model through convergent perturbation expansions,Commun. Math. Phys. 102:1–30 (1985).

    Google Scholar 

  153. K. Gawedzki and A. Kupiainen, Massless latticeφ 44 theory: Rigorous control of a renormalizable asymptotically free model,Commun. Math. Phys. 99:197–252 (1985).

    Google Scholar 

  154. K. Gawedzki and A. Kupiainen, Asymptotic freedom beyond perturbation theory, inCritical Phenomena, Random Systems, Gauge Theories [Les Houches 1984], K. Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1986), Part I, pp. 185–293.

    Google Scholar 

  155. D. Geman, Random fields and inverse problems in imaging, inEcole d'Eté de Probabilités de Saint-Flour XVIII-1988, Lecture Notes in Mathematics #1427, (Springer-Verlag, Berlin, 1990), pp. 116–193.

    Google Scholar 

  156. S. Geman, Hidden Markov models for image analysis, Lecture at Istituto per le Applicazioni del Calcolo, Rome (July 1990).

  157. S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,IEEE Trans. Pattern Anal. Machine Intell. 6:721–741 (1984).

    Google Scholar 

  158. H.-O. Georgii, Large deviations and maximum entropy principle for interacting random fields on ℤd,Ann. Prob., to appear.

  159. H.-O. Georgii, Two remarks on extremal equilibrium states,Commun. Math. Phys. 32:107–118 (1973).

    Google Scholar 

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

    Google Scholar 

  161. B. Gidas, A renormalization group approach to image processing problems,IEEE Trans. Pattern Anal. Mach. Intell. 11:164–180 (1989).

    Google Scholar 

  162. J. Glimm and A. Jaffe, Positivity of theϕ 43 Hamiltonian,Fortschr. Physik 21:327–376 (1973).

    Google Scholar 

  163. J. Glimm and A. Jaffe,Quantum Physics: A Functional Integral Point of View (Springer-Verlag, Berlin, 1981).

    Google Scholar 

  164. S. Goldstein, A note on specifications,Z. Wahrsch. verw. Geb. 46:45–51 (1978).

    Google Scholar 

  165. S. Goldstein, R. Kuik, J. L. Lebowitz, and C. Maes, From PCA's to equilibrium systems and back,Commun. Math. Phys. 125:71–79 (1989).

    Google Scholar 

  166. A. González-Arroyo, M. Okawa, and Y. Shimizu, Monte Carlo renormalization-group study of the four-dimensional Z2 gauge theory,Phys. Rev. Lett. 60:487–490 (1988).

    Google Scholar 

  167. A. González-Arroyo and J. Salas, Computing the couplings of Ising systems from Schwinger-Dyson equations,Phys. Lett. B 214:418–424 (1988).

    Google Scholar 

  168. A. González-Arroyo and J. Salas, Renormalization group flow of the two-dimensional Ising model atT< T c ,Phys. Lett. B 261:415–423 (1991).

    Google Scholar 

  169. M. Göpfert and G. Mack, Proof of confinement of static quarks in 3-dimensionalU(1) lattice gauge theory for all values of the coupling constant,Commun. Math. Phys. 82:545–606 (1982).

    Google Scholar 

  170. F. P. Greenleaf,Invariant Means on Topological Groups (Van Nostrand-Reinhold, New York, 1969).

    Google Scholar 

  171. R. B. Griffiths, Peierls' proof of spontaneous magnetization of a two-dimensional Ising ferromagnet,Phys. Rev. A136:437–439 (1964).

    Google Scholar 

  172. 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).

    Google Scholar 

  173. R. B. Griffiths, Phase diagrams and higher-order critical points,Phys. Rev. B 12:345–355 (1975).

    Google Scholar 

  174. R. B. Griffiths, Mathematical properties of renormalization-group transformations,Physica 106A:59–69 (1981).

    Google Scholar 

  175. 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 

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

    Google Scholar 

  177. R. B. Griffiths and D. Ruelle, Strict convexity (“continuity”) of the pressure in lattice systems,Commun. Math. Phys. 23:169–175 (1971).

    Google Scholar 

  178. C. Grillenberger and U. Krengel, On the spatial constant of superadditive set functions in ℝd, inErgodic Theory and Related Topics, H. Michel, ed. (Akademie-Verlag, Berlin, 1982).

    Google Scholar 

  179. P. Groeneboom, J. Oosterhoff, and F. H. Ruymgaart, Large deviation theorems for empirical probability measures,Ann. Prob. 7:553–586 (1979).

    Google Scholar 

  180. L. Gross, Absence of second-order phase transitions in the Dobrushin uniqueness region,J. Stat. Phys. 25:57–72 (1981).

    Google Scholar 

  181. L. Gross, Thermodynamics, statistical mechanics, and random fields, inEcole d'Eté de Probabilités de Saint-Flour X-1980, Lecture Notes in Mathematics #929, (Springer-Verlag, Berlin, 1982).

    Google Scholar 

  182. C. Gruber and A. Sütő, Phase diagrams of lattice systems of residual entropy,J. Stat. Phys. 42:113–142 (1988).

    Google Scholar 

  183. P. Hall and C. C. Heyde,Martingale Limit Theory and its Application (Academic Press, New York, 1980).

    Google Scholar 

  184. T. Hara, A rigorous control of logarithmic corrections in four-dimensionalϕ 4 spin systems. I. Trajectory of effective Hamiltonians,J. Stat. Phys. 47:57–98 (1987).

    Google Scholar 

  185. T. Hara, Mean-field critical behaviour for correlation length for percolation in high dimensions,Prob. Theory Related Fields 86:337–385 (1990).

    Google Scholar 

  186. T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions,Commun. Math. Phys. 128:333–391 (1990).

    Google Scholar 

  187. T. Hara and G. Slade, On the upper critical dimension of lattice trees and lattice animals,J. Stat. Phys. 59:1469–1510 (1990).

    Google Scholar 

  188. T. Hara and G. Slade, Critical behaviour of self-avoiding walk in five or more dimensions,Bull. Am. Math. Soc. 25:417–423 (1991).

    Google Scholar 

  189. T. Hara and G. Slade, The lace expansion for self-avoiding walk in five or more dimensions,Rev. Math. Phys. 4:235–327 (1992).

    Google Scholar 

  190. T. Hara and G. Slade, Self-avoiding walk in five or more dimensions. I. The critical behaviour,Commun. Math. Phys. 147:101–136 (1992).

    Google Scholar 

  191. T. Hara and H. Tasaki, A rigorous control of logarithmic corrections in four-dimensionalϕ 4 spin systems. II. Critical behavior of susceptibility and correlation length,J. Stat. Phys. 47:99–121 (1987).

    Google Scholar 

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

    Google Scholar 

  193. A. Hasenfratz, P. Hasenfratz, U. Heller, and F. Karsch, Improved Monte Carlo renormalization group methods,Phys. Lett. B 140:76–82 (1984).

    Google Scholar 

  194. T. Heath,A History of Greek Mathematics (Clarendon Press, Oxford, 1921), Vol. I, pp. 91–93, 380.

    Google Scholar 

  195. Y. Higuchi, On the absence of non-translationally invariant Gibbs states for the two-dimensional Ising system, inRandom Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory (Esztergom 1979), J. Fritz, J. L. Lebowitz, and D. Szász, eds. (North-Holland, Amsterdam, 1981).

    Google Scholar 

  196. Y. Higuchi and R. Lang, On the convergence of the Kadanoff transformation towards trivial fixed points,Z. Wahrsch. verw. Geb. 58:109–123 (1981).

    Google Scholar 

  197. P. Holický, R. Kotecký, and M. Zahradník, Rigid interfaces for lattice models at low temperatures,J. Stat. Phys. 50:755–812 (1988).

    Google Scholar 

  198. W. Holsztynski and J. Slawny, Peierls condition and the number of ground states,Commun. Math. Phys. 61:177–190 (1978).

    Google Scholar 

  199. P. J. Huber, The behavior of maximum likelihood estimates under nonstandard conditions, inProceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, L. M. Le Cam and J. Neyman, eds. (University of California Press, Berkeley, 1967), Vol. I, pp. 221–233.

    Google Scholar 

  200. O. Hudák, On the character of peculiarities in the position-space renormalization-group transformations,Phys. Lett. A 73:273–274 (1979).

    Google Scholar 

  201. N. M. Hugenholtz,C *-algebras and statistical mechanics, inProceedings of Symposia in Pure Mathematics, Volume 38, Part 2 (American Mathematical Society, Providence, Rhode Island, 1982), pp. 407–465.

    Google Scholar 

  202. N. M. Hugenholtz, On the inverse problem in statistical mechanics,Commun. Math. Phys. 85:27–38 (1982).

    Google Scholar 

  203. D. Iagolnitzer and B. Souillard, Random fields and limit theorems, inRandom Fields (Esztergom, 1979), J. Fritz, J. L. Lebowitz, and D. Szász, eds. (North-Holland, Amsterdam, 1981), Vol. II, pp. 573–591.

    Google Scholar 

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

    Google Scholar 

  205. J. Z. Imbrie, Phase diagrams and cluster expansions for low temperatureP(φ)2 models. I. The phase diagram,Commun. Math. Phys. 82:261–304 (1981).

    Google Scholar 

  206. J. Z. Imbrie, Phase diagrams and cluster expansions for low temperatureP(φ)2 models. II. The Schwinger function,Commun. Math. Phys. 82:305–343 (1981).

    Google Scholar 

  207. S. N. Isakov, Nonanalytic features of the first order phase transition in the Ising model,Commun. Math. Phys. 95:427–443 (1984).

    Google Scholar 

  208. R. B. Israel, High-temperature analyticity in classical lattice systems,Commun. Math. Phys. 50:245–257 (1976).

    Google Scholar 

  209. R. B. Israel,Convexity in the Theory of Lattice Gases (Princeton University Press, Princeton, New Jersey, 1979).

    Google Scholar 

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

    Google Scholar 

  211. R. B. Israel, Generic triviality of phase diagrams in spaces of long-range interactions,Commun. Math. Phys. 106:459–466 (1986).

    Google Scholar 

  212. R. B. Israel and R. R. Phelps, Some convexity questions arising in statistical mechanics,Math. Scand. 54:133–156 (1984).

    Google Scholar 

  213. L. P. Kadanoff and A. Houghton, Numerical evaluations of the critical properties of the two-dimensional Ising model,Phys. Rev. B 11:377–386 (1975).

    Google Scholar 

  214. J.-P. Kahane,Séries de Fourier Absolument Convergentes (Springer-Verlag, Berlin, 1970).

    Google Scholar 

  215. I. A. Kashapov, Justification of the renormalization-group method,Theor. Math. Phys. 42:184–186 (1980).

    Google Scholar 

  216. A. Katz,Principles of Statistical Mechanics (Freeman, San Francisco, 1967).

    Google Scholar 

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

    Google Scholar 

  218. T. Kennedy and E. H. Lieb, An itinerant electron model with crystalline or magnetic long-range order,Physica 138A:320–358 (1986).

    Google Scholar 

  219. J. C. Kieffer, A counterexample to Perez's generalization of the Shannon-McMillan theorem,Ann. Prob. 1:362–364 (1973); Correction,Ann. Prob. 4:153–154 (1976).

    Google Scholar 

  220. W. Klein, D. J. Wallace, and R. K. P. Zia, Essential singularities at first-order phase transitions,Phys. Rev. Lett. 37:639–642 (1976).

    Google Scholar 

  221. H. Koch and P. Wittwer, A non-Gaussian renormalization group fixed point for hierarchical scalar lattice field theories,Commun. Math. Phys. 106:495–532 (1986).

    Google Scholar 

  222. H. Koch and P. Wittwer, On the renormalization group transformation for scalar hierarchical models,Commun. Math. Phys. 138:537–568 (1991).

    Google Scholar 

  223. J. M. Kosterlitz, The critical properties of the two-dimensionalxy model,J. Phys. C 7:1046–1060 (1974).

    Google Scholar 

  224. F. Koukiou, D. Petritis, and M. Zahradnik, Extension of the Pirogov-Sinai theory to a class of quasiperiodic interactions,Commun. Math. Phys. 118:365–383 (1988).

    Google Scholar 

  225. O. K. Kozlov, Gibbs description of a system of random variables,Probl. Inform. Transmission 10:258–265 (1974).

    Google Scholar 

  226. U. Krengel,Ergodic Theorems (de Gruyter, Berlin, 1985).

    Google Scholar 

  227. K. Krickeberg,Probability Theory (Addison-Wesley, Reading, Massachusetts, 1965).

    Google Scholar 

  228. S. Kullback,Information Theory and Statistics (Wiley, New York, 1959).

    Google Scholar 

  229. S. Kullback and R. A. Leibler, On information and sufficiency,Ann. Math. Stat. 22:79–86 (1951).

    Google Scholar 

  230. H. Künsch, Thermodynamics and statistical analysis of Gaussian random fields,Z. Wahrsch. verw. Geb. 58:407–421 (1981).

    Google Scholar 

  231. H. Künsch, Decay of correlations under Dobrushin's uniqueness condition and its applications,Commun. Math. Phys. 84:207–222 (1982).

    Google Scholar 

  232. H. Künsch, Non-reversible stationary measures for infinite interacting particle systems,Z. Wahrsch. verw. Geb. 66:407–424 (1984).

    Google Scholar 

  233. O. E. Lanford III, Entropy and equilibrium states in classical statistical mechanics, inStatistical Mechanics and Mathematical Problems (Battelle Seattle Rencontres 1971), Lecture Notes in Physics #20 (Springer-Verlag, Berlin, 1973), pp. 1–113.

    Google Scholar 

  234. O. E. Lanford III and D. W. Robinson, Statistical mechanics of quantum spin systems. III,Commun. Math. Phys. 9:327–338 (1968).

    Google Scholar 

  235. O. E. Lanford III and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics,Commun. Math. Phys. 13:194–215 (1969).

    Google Scholar 

  236. C. B. Lang, Renormalization study of compactU(1) lattice gauge theory,Nucl. Phys. B 280[FS18]:255–275 (1987).

    Google Scholar 

  237. S. Lang,Introduction to Diophantine Approximations (Addison-Wesley, Reading, 1966).

    Google Scholar 

  238. I. D. Lawrie, Tricritical scaling and renormalisation ofφ ∂ operators in scalar systems near four dimensions,J. Phys. A 12:919–940 (1979).

    Google Scholar 

  239. V. A. Lebesgue, Sur l'équation indéterminéex 5 +y 5 =az 5,J. Math. Pures Appl. 8:49–70 (1843).

    Google Scholar 

  240. J. L. Lebowitz, Coexistence of phases in Ising ferromagnets,J. Stat. Phys. 16:463–476 (1977).

    Google Scholar 

  241. J. L. Lebowitz, Number of phases in one component ferromagnets, inMathematical Problems in Theoretical Physics, Lecture Notes in Physics #80, G. Dell'Antonio, S. Doplicher, and G. Jona-Lasinio, eds. (Springer-Verlag, Berlin, 1978).

    Google Scholar 

  242. J. L. Lebowitz, Microscopic origin of hydrodynamic equations: Derivation and consequences,Physica 140A:232–239 (1986).

    Google Scholar 

  243. J. L. Lebowitz and C. Maes, The effect of an external field on an interface, entropic repulsion,J. Stat. Phys. 46:39–49 (1987).

    Google Scholar 

  244. J. L. Lebowitz, C. Maes, and E. R. Speer, Statistical mechanics of probabilistic cellular automata,J. Stat. Phys. 59:117–170 (1990).

    Google Scholar 

  245. J. L. Lebowitz and O. Penrose, Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapor transition,J. Math. Phys. 7:98–113 (1966).

    Google Scholar 

  246. J. L. Lebowitz and O. Penrose, Analytic and clustering properties of thermodynamic functions and distribution functions for classical lattice and continuous systems,Commun. Math. Phys. 11:99–124 (1968).

    Google Scholar 

  247. J. L. Lebowitz and O. Penrose, Divergent susceptibility of isotropic ferromagnets,Phys. Rev. Lett. 35:549–551 (1975).

    Google Scholar 

  248. J. L. Lebowitz, M. K. Phani, and D. F. Styer, Phase diagram of Cu-Au-type alloys,J. Stat. Phys. 38:413–431 (1985).

    Google Scholar 

  249. J. L. Lebowitz and E. Presutti, Statistical mechanics of unbounded spin systems,Commun. Math. Phys. 50:195–218 (1976).

    Google Scholar 

  250. J. L. Lebowitz and R. H. Schonmann, Pseudo-free energies and large deviations for non-Gibbsian FKG measures,Prob. Theory Related Fields 77:49–64 (1988).

    Google Scholar 

  251. D. H. Lehmer [Review of ref. 76],Math. Rev. 16:903 (1955).

    Google Scholar 

  252. B. G. Leroux, Maximum-likelihood estimation for hidden Markov models,Stoch. Proc. Appl. 40:127–143 (1992).

    Google Scholar 

  253. A. L. Lewis, Lattice renormalization group and thermodynamic limit,Phys. Rev. B 16:1249–1252 (1977).

    Google Scholar 

  254. E. H. Lieb and A. D. Sokal, A general Lee-Yang theorem for one-component and multicomponent ferromagnets,Commun. Math. Phys. 80:153–179 (1981).

    Google Scholar 

  255. T. M. Liggett,Interacting Particle Systems (Springer-Verlag, Berlin, 1985).

    Google Scholar 

  256. J. Lindenstrauss, G. Olsen, and Y. Sternfeld, The Poulsen simplex,Ann. Inst. Fourier (Grenoble)28:91–114 (1978).

    Google Scholar 

  257. J. Lörinczi and M. Winnink, Some remarks on almost Gibbs states, inProceedings of the NATO Advanced Studies Institute Workshop on Cellular Automata and Cooperative Systems [Les Houches 1992], N. Boccara, E. Goles, S. Martinez, and P. Picco, eds. (Kluwer, Dordrecht, 1993).

    Google Scholar 

  258. S. K. Ma,Modern Theory of Critical Phenomena (Benjamin, Reading, Massachusetts, 1976).

    Google Scholar 

  259. C. Maes and F. Redig, Anisotropic perturbations of the simple symmetric exclusion process: Long range correlations,J. Phys. I (Paris)1:669–684 (1991).

    Google Scholar 

  260. C. Maes and K. van de Velde, Defining relative energies for the projected Ising measure,Helv. Phys. Acta 65:1055–1068 (1992).

    Google Scholar 

  261. C. Maes and K. van de Velde, The interaction potential of the stationary measure of a high-noise spinflip process,J. Math. Phys. 34:3030–3038 (1993).

    Google Scholar 

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

    Google Scholar 

  263. F. Martinelli and E. Scoppola, A simple stochastic cluster dynamics: Rigorous results,J. Phys. A 24:3135–3157 (1991).

    Google Scholar 

  264. D. G. Martirosyan, Uniqueness of Gibbs states in lattice models with one ground state,Theor. Math. Phys. 63:511–518 (1985).

    Google Scholar 

  265. A. E. Mazel and Yu. M. Suhov, Random surfaces with two-sided constraints: An application of the theory of dominant ground states,J. Stat. Phys. 64:111–134 (1991).

    Google Scholar 

  266. A. Messager, S. Miracle-Solé, and C.-E. Pfister, Correlation inequalities and uniqueness of the equilibrium state for the plane rotator ferromagnetic model,Commun. Math. Phys. 58:19–30 (1978).

    Google Scholar 

  267. J. Miekisz, The global minimum of energy is not always a sum of local minima—A note on frustration,J. Stat. Phys. 71:425–434 (1993).

    Google Scholar 

  268. J. Miekisz, Classical lattice gas model with a unique nondegenerate but unstable periodic ground state configuration,Commun. Math. Phys. 111:533–538 (1987).

    Google Scholar 

  269. M. Mignotte and M. Waldschmidt, Linear forms in two logarithms and Schneider's method. II,Acta Arith. 53:251–287 (1989).

    Google Scholar 

  270. L. J. Mordell,Diophantine Equations (Academic Press, New York, 1969).

    Google Scholar 

  271. J. Moulin-Ollagnier, Théorème ergodique presque sous-additif et convergence en moyenne de l'information,Ann. Inst. Henri Poincaré B 19:257–266 (1983).

    Google Scholar 

  272. J. Moulin-Ollagnier and D. Pinchon, Mesures de Gibbs invariantes et mesures d'équilibre,Z. Wahrsch. verw. Geb. 55:11–23 (1981).

    Google Scholar 

  273. J. Moulin-Ollagnier and D. Pinchon, Filtre moyennant et valeurs moyennes de capacités invariantes,Bull. Soc. Math. Fr. 110:259–277 (1982).

    Google Scholar 

  274. J. Moussouris, Gibbs and Markov random systems with constraints,J. Stat. Phys. 10:11–33 (1974).

    Google Scholar 

  275. C. Mugler,Platon et la Recherche Mathématique de son Époque (P. H. Heitz, Strasbourg-Zürich, 1948), pp. 226–236.

    Google Scholar 

  276. D. R. Nelson, Coexistence-curve singularities in isotropic ferromagnets,Phys. Rev. B 13:2222–2230 (1976).

    Google Scholar 

  277. J. Neveu,Bases Mathématiques du Calcul des Probabilités, 2nd éd. (Masson, Paris, 1980) [English translation of first edition:Mathematical Foundations of the Calculus of Probability (Holden-Day, San Francisco, 1965)].

    Google Scholar 

  278. C. M. Newman, Normal fluctuations and the FKG inequalities,Commun. Math. Phys. 74:119–128 (1980).

    Google Scholar 

  279. C. M. Newman, A general central limit theorem for FKG systems,Commun. Math. Phys. 91:75–80 (1983).

    Google Scholar 

  280. C. M. Newman, Private communication (1984).

  281. Th. Niemeijer and J. M. J. van Leeuwen, Wilson theory for spin systems on a triangular lattice,Phys. Rev. Lett. 31:1411–1414 (1973).

    Google Scholar 

  282. Th. Niemeijer and J. M. J. van Leeuwen, Renormalization theory for Ising-like spin systems, inPhase Transitions and Critical Phenomena, Vol. 6, C. Domb and M. S. Green, eds. (Academic Press, New York, 1976).

    Google Scholar 

  283. Th. Niemeyer and J. M. J. van Leeuwen, Wilson theory for 2-dimensional Ising spin systems,Physica 71:17–40 (1974).

    Google Scholar 

  284. B. Nienhuis, A. N. Berker, E. K. Riedel, and M. Schick, First- and second-order phase transitions in Potts models: Renormalization-group solution,Phys. Rev. Lett. 43:737–740 (1979).

    Google Scholar 

  285. B. Nienhuis and M. Nauenberg, First-order phase transitions in renormalization-group theory,Phys. Rev. Lett. 35:477–479 (1975).

    Google Scholar 

  286. G. L. O'Brien, Scaling transformations for {0, 1}-valued sequences,Z. Wahrsch. verw. Geb. 53:35–49 (1980).

    Google Scholar 

  287. S. Olla, Large deviations for almost Markovian processes,Prob. Theory Related Fields 76:395–409 (1987).

    Google Scholar 

  288. S. Olla, Large deviations for Gibbs random fields,Prob. Theory Related Fields 77:343–357 (1988).

    Google Scholar 

  289. G. H. Olsen, On simplices and the Poulsen simplex, inFunctional Analysis: Surveys and Recent Results II [Proceedings of the Conference on Functional Analysis, Paderborn, Germany, 1979] (North-Holland, Amsterdam, 1980), pp. 31–52.

    Google Scholar 

  290. J. C. Oxtoby,Measure and Category (Springer-Verlag, Berlin, 1971).

    Google Scholar 

  291. Y. M. Park, Cluster expansion for classical and quantum lattice systems,J. Stat. Phys. 27:553–576 (1982).

    Google Scholar 

  292. Y. M. Park, Extension of Pirogov-Sinai theory of phase transitions to infinite range interactions. I. Cluster expansion,Commun. Math. Phys. 114:187–218 (1988).

    Google Scholar 

  293. Y. M. Park, Extension of Pirogov-Sinai theory of phase transitions to infinite range interactions. II. Phase diagram,Commun. Math. Phys. 114:219–241 (1988).

    Google Scholar 

  294. K. R. Parthasarathy,Probability Measures on Metric Spaces (Academic Press, New York, 1967).

    Google Scholar 

  295. E. A. Pecherski, The Peierls condition (GPS condition) is not always satisfied,Selecta Math. Sov. 3:87–91 (1983/1984).

    Google Scholar 

  296. R. Peierls, Ising's model of ferromagnetism,Proc. Camb. Phil. Soc. 32:477–481 (1936).

    Google Scholar 

  297. P. Pfeuty and G. Toulouse,Introduction to the Renormalization Group and to Critical Phenomena (Wiley, New York, 1977).

    Google Scholar 

  298. R. R. Phelps,Lectures on Choquet's Theorem (Van Nostrand, Princeton, New Jersey, 1966).

    Google Scholar 

  299. R. R. Phelps, Generic Fréchet differentiability of the pressure in certain lattice systems,Commun. Math. Phys. 91:557–562 (1983).

    Google Scholar 

  300. S. A. Pirogov, Coexistence of phases in a multicomponent lattice liquid with complex thermodynamic parameters,Theor. Math. Phys. 66:218–221 (1986).

    Google Scholar 

  301. S. A. Pirogov and Ya. G. Sinai, Phase diagrams of classical lattice systems,Theor. Math. Phys. 25:1185–1192 (1976).

    Google Scholar 

  302. S. A. Pirogov and Ya. G. Sinai, Phase diagrams of classical lattice systems. Continuation,Theor. Math. Phys. 26:39–49 (1976).

    Google Scholar 

  303. C. Prakash, High-temperature differentiability of lattice Gibbs states by Dobrushin uniqueness techniques,J. Stat. Phys. 31:169–228 (1983).

    Google Scholar 

  304. C. Preston,Random Fields (Springer-Verlag, Berlin, 1976).

    Google Scholar 

  305. C. Preston, Construction of specifications, inQuantum Fields, Algebras, Processes, Lecture Notes in Mathematics #534, L. Streit, ed. (Springer-Verlag, Berlin, 1980).

    Google Scholar 

  306. Proclus,Commentaire sur la République [Translation and notes by A. J. Festugière] (Librairie Philosophique J. Vrin, Paris, 1970), Vol. II, pp. 133–135.

    Google Scholar 

  307. L. R. Rabiner, A tutorial on hidden Markov models and selected applications in speech recognition,Proc. IEEE 77:257–286 (1989).

    Google Scholar 

  308. C. Radin, Low temperature and the origin of crystalline symmetry,Int. J. Mod. Phys. B 1:1157–1191 (1987).

    Google Scholar 

  309. C. Radin, Disordered ground states of classical lattice models,Rev. Math. Phys. 3:125–135 (1991).

    Google Scholar 

  310. C. Rebbi and R. H. Swendsen, Monte Carlo renormalization-group studies ofq-state Potts models in two dimensions,Phys. Rev. B 21:4094–4107 (1980).

    Google Scholar 

  311. M. Reed and B. Simon,Methods of Modern Mathematical Physics I: Functional Analysis (Academic Press, New York, 1972).

    Google Scholar 

  312. L. E. Reichl,A Modern Course in Statistical Physics (University of Texas Press, Austin, Texas, 1980).

    Google Scholar 

  313. P. Ribenboim,The Book of Prime Number Records, 2nd ed. (Springer-Verlag, Berlin, 1989).

    Google Scholar 

  314. H. L. Royden,Real Analysis, 2nd ed. (Macmillan, New York, 1968).

    Google Scholar 

  315. W. Rudin,Functional Analysis (McGraw-Hill, New York, 1973).

    Google Scholar 

  316. D. Ruelle, Some remarks on the ground state of infinite systems in statistical mechanics,Commun. Math. Phys. 11:339–345 (1969).

    Google Scholar 

  317. D. Ruelle,Statistical Mechanics: Rigorous Results (Benjamin, Reading, Massachusetts, 1969).

    Google Scholar 

  318. D. Ruelle,Thermodynamic Formalism (Addison-Wesley, Reading, Massachusetts, 1978).

    Google Scholar 

  319. J. Salas, Private communication (1991).

  320. S. Sarbach and M. E. Fisher, Tricritical scaling in the spherical model limit,J. Appl. Phys. 49:1350–1352 (1978).

    Google Scholar 

  321. S. Sarbach and M. E. Fisher, Tricriticality and the failure of scaling in the manycomponent limit,Phys. Rev. B 18:2350–2363 (1978).

    Google Scholar 

  322. S. Sarbach and M. E. Fisher, Tricritical coexistence in three dimensions: The multicomponent limit,Phys. Rev. B 20:2797–2817 (1979).

    Google Scholar 

  323. H. H. Schaefer,Topological Vector Spaces (Springer-Verlag, Berlin, 1980).

    Google Scholar 

  324. A. G. Schlijper, Tiling problems and undecidability in the cluster variation method,J. Stat. Phys. 50:689–714 (1988).

    Google Scholar 

  325. W. Schmidt,Diophantine Approximation, Lecture Notes in Mathematics #785 (Springer-Verlag, Berlin, 1980).

    Google Scholar 

  326. R. H. Schonmann, Second order large deviation estimates for ferromagnetic systems in the phase coexistence region,Commun. Math. Phys. 112:409–422 (1987).

    Google Scholar 

  327. R. H. Schonmann, Projections of Gibbs measures may be non-Gibbsian,Commun. Math. Phys. 124:1–7 (1989).

    Google Scholar 

  328. R. Schrader, Ground states in classical lattice systems with hard core,Commun. Math. Phys. 16:247–264 (1970).

    Google Scholar 

  329. L. Schwartz,Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures (Tata Institute of Fundamental Research and Oxford University Press, Oxford, 1973).

    Google Scholar 

  330. S. H. Shenker and J. Tobochnik, Monte Carlo renormalization-group analysis of the classical Heisenberg model in two dimensions,Phys. Rev. B 22:4462–4472 (1980).

    Google Scholar 

  331. S. B. Shlosman, Uniqueness and half-space nonuniqueness of Gibbs states in Czech models,Theor. Math. Phys. 66:284–293 (1986).

    Google Scholar 

  332. S. B. Shlosman, Gaussian behavior of the critical Ising model in dimensiond>4,Sov. Phys. Doklady 33:905–906 (1988).

    Google Scholar 

  333. S. B. Shlosman, Relations among the cumulants of random fields with attraction,Theor. Prob. Appl. 33:645–655 (1989).

    Google Scholar 

  334. S. D. Silvey,Statistical Inference (Chapman and Hall, London, 1975).

    Google Scholar 

  335. B. Simon and A. D. Sokal, Rigorous entropy-energy arguments,J. Stat. Phys. 25:679–694 (1981); Addendum,J. Stat. Phys. 29:155 (1982).

    Google Scholar 

  336. Ya. G. Sinai, Self-similar probability distributions,Theor. Prob. Appl. 21:64–80 (1976).

    Google Scholar 

  337. Ya. G. Sinai,Theory of Phase Transitions: Rigorous Results (Pergamon Press, Oxford, 1982).

    Google Scholar 

  338. J. Slawny, Low-temperature properties of classical lattice systems: Phase transitions and phase diagrams, inPhase Transitions and Critical Phenomena, Vol. 11, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1985).

    Google Scholar 

  339. A. D. Sokal, Existence of compatible families of proper regular conditional probabilities,Z. Wahrsch. verw. Geb. 56:537–548 (1981).

    Google Scholar 

  340. A. D. Sokal, Unpublished (1982).

  341. A. D. Sokal, More surprises in the general theory of lattice systems,Commun. Math. Phys. 86:327–336 (1982).

    Google Scholar 

  342. A. D. Sokal, Subadditive set functions on a discrete amenable group, Unpublished manuscript (1984).

  343. A. Stella, Singularities in renormalization group transformations,Physica 108A:211–220 (1981).

    Google Scholar 

  344. K. Subbarao, Renormalization group for Ising spins on a finite lattice,Phys. Rev. B 11:1165–1168 (1975).

    Google Scholar 

  345. W. G. Sullivan, Potentials for almost Markovian random fields,Commun. Math. Phys. 33:61–74 (1973).

    Google Scholar 

  346. R. H. Swendsen, Monte Carlo renormalization, inReal-Space Renormalization, T. W. Burkhardt and J. M. J. van Leeuwen, eds. (Springer-Verlag, New York, 1982), pp. 57–86.

    Google Scholar 

  347. R. H. Swendsen, Monte-Carlo renormalization group, inPhase Transitions (Cargèse 1980), M. Lévy, J.-C. LeGuillou, and J. Zinn-Justin, eds. (Plenum Press, New York, 1982), pp. 395–422.

    Google Scholar 

  348. R. H. Swendsen, Monte Carlo calculation of renormalized coupling parameters. I.d=2 Ising model,Phys. Rev. B 30:3866–3874 (1984).

    Google Scholar 

  349. R. H. Swendsen and J.-S. Wang, Non-universal critical dynamics in Monte Carlo simulations,Phys. Rev. Lett. 58:86–88 (1987).

    Google Scholar 

  350. G. S. Sylvester, Inequalities for continuous spin Ising ferromagnets,J. Stat. Phys. 15:327–341 (1976).

    Google Scholar 

  351. I. Syozi, Transformation of Ising models, inPhase Transitions and Critical Phenomena, Vol. 1, C. Domb and M. S. Green, eds. (Academic Press, New York, 1972).

    Google Scholar 

  352. P. Tannery, Review of H. Konen, Geschichte der Gleichungt 2−Du 2=1,Bull. Sci. Math. 27:47–51 (1903).

    Google Scholar 

  353. Théon de Smyrne,Exposition des Connaissances Mathématiques Utiles pour la Lecture de Platon, [translated by J. Dupuis] (Hachette, Paris, 1892), pp. 71–75.

    Google Scholar 

  354. C. J. Thompson,Mathematical Statistical Mechanics (Princeton University Press, Princeton, New Jersey, 1979).

    Google Scholar 

  355. C. J. Thompson,Classical Equilibrium Statistical Mechanics (Clarendon Press, Oxford, 1988).

    Google Scholar 

  356. A. L. Toom, Stable and attractive trajectories in multicomponent systems, inMulticomponent Random Systems, R. L. Dobrushin and Ya. G. Sinai, eds. (Marcel Dekker, New York, 1980), pp. 549–575.

    Google Scholar 

  357. H. van Beijeren, Interface sharpness in the Ising system,Commun. Math. Phys. 40:1–6 (1975).

    Google Scholar 

  358. A. C. D. van Enter, A note on the stability of phase diagrams in lattice systems,Commun. Math. Phys. 79:25–32 (1981).

    Google Scholar 

  359. A. C. D. van Enter, Instability of phase diagrams for a class of “irrelevant” perturbations,Phys. Rev. B 26:1336–1339 (1982).

    Google Scholar 

  360. A. C. D. van Enter and R. Fernández, A remark on different norms and analyticity for many-particle interactions,J. Stat. Phys. 56:965–972 (1989).

    Google Scholar 

  361. A. C. D. van Enter, R. Fernández, and A. D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations,Nucl. Phys. B (Proc. Suppl.) 20:48–52 (1991).

    Google Scholar 

  362. 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 

  363. A. C. D. van Enter, R. Fernández, and A. D. Sokal, Non-Gibbsian states, inProceedings of the 1992 Prague Workshop on Phase Transitions, R. Kotecký, ed. (World Scientific, Singapore, 1993).

    Google Scholar 

  364. A. C. D. van Enter, R. Fernández, and A. D. Sokal, Non-Gibbsian states for renormalization-group transformations and beyond, inProceedings of the NATO Advanced Studies Institute Workshop on Cellular Automata and Cooperative Systems [Les Houches 1992, N. Boccara, E. Goles, S. Martinez, and P. Picco, eds. (Kluwer, Dordrecht, 1993).

    Google Scholar 

  365. A. C. D. van Enter and J. Miekisz, Breaking of periodicity at positive temperatures,Commun. Math. Phys. 134:647–651 (1990).

    Google Scholar 

  366. J. M. J. van Leeuwen, Singularities in the critical surface and universality for Ising-like spin systems,Phys. Rev. Lett. 34:1056–1058 (1975).

    Google Scholar 

  367. V. S. Varadarajan, Measures on topological spaces [in Russian],Mat. Sbornik N.S. 55(97):35–100 (1961) [English translation],Am. Math. Soc. Transl. Ser. 2 48:161–228 (1965)].

    Google Scholar 

  368. S. R. S. Varadhan, Private communication (1984).

  369. S. R. S. Varadhan,Large Deviations and Applications (SIAM, Philadelphia, Pennsylvania, 1984).

    Google Scholar 

  370. J. Voigt, Stochastic operators, information and entropy,Commun. Math. Phys. 81:31–38 (1981).

    Google Scholar 

  371. J.-S. Wang and J. L. Lebowitz, Phase transitions and universality in nonequilibrium steady states of stochastic Ising models,J. Stat. Phys. 51:893–906 (1988).

    Google Scholar 

  372. F. J. Wegner and E. K. Riedel, Logarithmic corrections to the molecular-field behavior of critical and tricritical systems,Phys. Rev. B 7:248–256 (1973).

    Google Scholar 

  373. A. S. Wightman, Convexity and the notion of equilibrium state in thermodynamics and statistical mechanics. Introduction to R. B. Israel,Convexity in the Theory of Lattice Gases (Princeton University Press, Princeton, New Jersey, 1979).

    Google Scholar 

  374. K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem,Rev. Mod. Phys. 47:773–840 (1975).

    Google Scholar 

  375. K. G. Wilson and J. Kogut, The renormalization group and the ε-expansion,Phys. Rep. 12C:75–200 (1974).

    Google Scholar 

  376. M. Zahradník, An alternate version of Pirogov-Sinai theory,Commun. Math. Phys. 93:559–581 (1984).

    Google Scholar 

  377. M. Zahradník, Low temperature continuous spin Gibbs states on a lattice and the interfaces between them—A Pirogov-Sinai type approach, inStatistical Mechanics and Field Theory: Mathematical Aspects (Proceedings, Groningen 1985), Lecture Notes in Physics #257, T. C. Dorlas, N. M. Hugenholtz, and M. Winnink, eds. (Springer-Verlag, Berlin, 1986).

    Google Scholar 

  378. M. Zahradník, Analyticity of low-temperature phase diagrams of lattice spin models,J. Stat. Phys. 47:725–755 (1987).

    Google Scholar 

  379. M. Zahradník, Phase diagrams of lattice spin models. Cours de Troisième Cycle de la Physique en Suisse Romande, Lausanne (May 1987).

  380. M. Zahradník, Low temperature phase diagrams of lattice models with random impurities, Preprint (1988).

  381. M. Zahradník, Private communication (1990).

  382. F. Igloi and C. Vanderzande, Renormalisation group study of the (2 + 1) dimensional Potts model,Physica A135:347–358 (1986).

    Google Scholar 

  383. K. A. Ribet, Wiles proves Taniyama's conjecture: Fermat's last theorem follows,Notices Amer. Math. Soc. 40:575–576 (1993).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, Rijksuniversiteit Groningen, 9747, AG Groningen, The Netherlands

    Aernout C. D. van Enter

  2. Institut de Physique Théorique, Ecole Polytechnique Fédérale de Lausanne, PHB-Ecublens, CH-1015, Lausanne, Switzerland

    Roberto Fernández

  3. Department of Physics, New York University, 10003, New York, New York

    Alan D. Sokal

Authors
  1. Aernout C. D. van Enter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto Fernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alan D. Sokal
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

van Enter, A.C.D., Fernández, R. & Sokal, A.D. Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory. J Stat Phys 72, 879–1167 (1993). https://doi.org/10.1007/BF01048183

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01048183

Key words

Search

Navigation