Skip to main content
SpringerLink
Log in

Random Spin Systems with Long-Range Interactions

  • Chapter
Mathematical Aspects of Spin Glasses and Neural Networks

Part of the book series: Progress in Probability ((PRPR,volume 41))

Abstract

We present the state of the matter for random spin systems with long-range interactions.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Binder, K and Young, A.P., Spin glasses: Experimental facts, theoretical concepts and open questions, Rev. Mod. Phys. 58 (1986) 801.

    Article  Google Scholar 

  2. Bishop, E. and Phelps R.R., The support Junctionals of a convex set AMS Proceedings of Symposia in Pure Mathematics, Vol. 7 pp. 25–35, American Mathematical Society, Providence, RI, 1963.

    Google Scholar 

  3. Bonato, C.A. and Campanino, M., Absence of symmetry breaking for Systems of Rotors with Random Interactions, J. Stat. Phys. 54 (1989) 81–88.

    Article  MathSciNet  MATH  Google Scholar 

  4. Borgs, C. and Imbrie J.Z., A Unified Approach to Phase Diagrams in Field Theory and Statistical Mechanics, Commun. Math. Phys. 123 (1989) 305–328.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bricmont, J. and Slawny, J., Phase Transitions in Systems with a Finite Number of Dominant Ground States, J. Stat. Phys. 54 (1989) 89–161.

    Article  MathSciNet  Google Scholar 

  6. Bricmont, J., Kuroda, K. and Lebowitz, J.L., The Structure of Gibbs States and Phase Coexistence for Non-Symmetric Continuum Widom Rolinson Models, Z. Wahrsch. verw. Geb. 67 (1984) 121–138.

    Article  MathSciNet  MATH  Google Scholar 

  7. Brickmont, J. and Kupiainen, A., Phase transition in the 3d Random Field Ising Model, Commun. Math. Phys. 116 (1988) 539–572.

    Article  Google Scholar 

  8. Comets, F., Large Deviation Estimates for a Conditional Probability Distribution. Applications to Random Interaction Gibbs Measures, Probab. Theor. Rel. Fields 80 (1989) 407–432.

    Article  MathSciNet  MATH  Google Scholar 

  9. Chowdhury, D., Spin glasses and other frustrated systems World Scientific, Singapore, 1986.

    Google Scholar 

  10. Campanino, M., Olivieri, E. and van Enter, A.C.D., One-dimensional spin glasses with potential decay r −(1+ε)Absence of phase transitions and cluster properties, Commun. Math. Phys. 108 (1987) 241–255.

    Article  MATH  Google Scholar 

  11. Dobrushin, R.L. and Martirosyan, M.R., Nonfinite Perturbations of Gibbs Fields, Theor. Math. Phys. 74 (1988) 10–20.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Edwards, S.F. and Anderson, P.W, Theory of spin glasses, J. Phys. F (Metal. Phys.) 5 (1975) 965–974.

    Article  Google Scholar 

  14. van Enter, A.C.D., A Note on the Stability of Phase Diagrams in Lattice Systems, Commun. Math. Phys. 79 (1981) 25–32.

    Article  Google Scholar 

  15. van Enter, A.C.D., Instability of phase diagrams for a class of “irrelevant” perturbations, Phys. Rev. B26 (1982) 1336–1339.

    Google Scholar 

  16. van Enter, A.C.D., and van Hemmen, J.L., The thermodynamic limit for long-range random systems, J. Stat. Phys. 32 (1983) 141–152.

    Article  Google Scholar 

  17. van Enter, A.C.D., and van Hemmen, J.L., Absence of phase transitions in certain one-dimensional long-range random systems, J. Stat. Phys. 39 (1985) 1–14.

    Article  MATH  Google Scholar 

  18. van Enter, A.C.D., and Fröhlich, J., Absence of symmetry breaking for N-vector spin glass models in two dimensions, Commun. Math. Phys. 98 (1985) 425–432.

    Article  Google Scholar 

  19. van Enter, A.C.D., Bounds on correlation decay for long-range vector spin-glasses, J. Stat. Phys. 11 (1985) 315.

    Article  Google Scholar 

  20. van Enter, A.C.D., One-dimensional spin glasses; uniqueness and cluster properties, J. Phys. A21 (1988) 1781–1786.

    Google Scholar 

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

    Article  MATH  Google Scholar 

  22. van Enter, A.C.D., and Zegarlinski, B., A remark on differentiability of the pressure functional, Rev. Math. Phys. 7 (1995) 959–977.

    Article  MathSciNet  MATH  Google Scholar 

  23. van Enter, A.C.D., and Zegarlinski, B., Non-periodic long-range order for one-dimensional pair interactions, J. Phys. A30 (1997) 501–505.

    Google Scholar 

  24. Figotin, A.L., and Pastur, L.A., Theor. Mat. Fiz. 35 (1978) 193.

    MathSciNet  Google Scholar 

  25. Föllmer, H., Phase transition and Martin boundary 465 pp. 305–317, Lecture Notes in Mathematics in Seminare de Probabilities IX, Strasbourg 1975, Springer-Verlag.

    Google Scholar 

  26. Fröhlich, J., Mathematical aspects of the physics of disordered systems. pp. 725–894 in Vol. II, Critical Phenomena, Random Systems and Gauge Theories Osterwalder, K. and Stora, R., eds., North-Holland, 1986.

    Google Scholar 

  27. Fröhlich, J., and Zegarlinski, B., The disordered phase of long-range Ising spin glasses, Europhys. Lett. 2 (1986) 53–60.

    Article  Google Scholar 

  28. Fröhlich, J., and Zegarlinski, B., The high-temperature phase of long-range spin glasses, Commun. Math. Phys. 110 (1987) 121–155.

    Article  Google Scholar 

  29. Fröhlich, J., and Zegarlinski, B., Some comments on the Sherrington–Kirkpatric Model of spin glasses, Commun. Math. Phys. 112 (1987) 553–566.

    Article  Google Scholar 

  30. Fröhlich, J., and Zegarlinski, B., Spin glasses and other lattice systems with long-range interactions, Commun. Math. Phys. 120 (1989) 665–688.

    Article  MATH  Google Scholar 

  31. Georgii, H.O., Gibbs Measures and Phase Transitions Walter-de-Gruyter, Berlin, 1988.

    Book  MATH  Google Scholar 

  32. Goulart-Rosa S., The thermodynamic limit of quenched free energy of magnetic systems with random interactionsJ. Phys. A (Math. Gen.) 15 (1982) L51–54.

    Article  MathSciNet  Google Scholar 

  33. Gandolfi, A., Newman, C.M., and Stein, D.L., Exotic states in long-range spin glasses, Commun. Math. Phys. 157. (1993) 371–387.

    Article  MathSciNet  MATH  Google Scholar 

  34. Ginibre, J., Grossman, A., and Ruelle D., Condensation of lattice gases, J. Stat. Phys. 3 (1966) 187–193.

    Google Scholar 

  35. Gruber C. and Süto A., Phase diagrams of lattice systems with residual entropy, J. Stat. Phys. 57 (1988) 113–142.

    Article  Google Scholar 

  36. Gruber C., Süto A., Lemberger P., Phase diagrams of lattice systems with residual entropy, II. Low temperature expansion, J. Stat. Phys. 57 (1988) 113–142.

    Article  Google Scholar 

  37. Guionnet A. and Zegarlinski B., Decay to equilibrium in random spin systems on a lattice, Commun. Math. Phys., 181. (1996) 703–732.

    Article  MathSciNet  MATH  Google Scholar 

  38. Guionnet A. and Zegarlinski B., Decay to equilibrium in random spin systems on a lattice, II, J. Stat. Phys. 86. (1997) 899–904.

    Article  MathSciNet  MATH  Google Scholar 

  39. van Hemmen, J.L. and Morgenstern, I., eds., Heidelberg Colloquium on Spin Glasses LNP 192 Springer-Verlag, 1983.

    Google Scholar 

  40. Imbrie J.Z., Phase diagrams and cluster expansions for low temperature P(φ)2 models, ICommun. Math. Phys. 82 (1981) 261–304;

    Article  MathSciNet  Google Scholar 

  41. Imbrie J.Z. Phase diagrams and cluster expansions for low temperature P(φ)2 models, II, Commun. Math. Phys.. 82 (1981) 305–343.

    Article  MathSciNet  Google Scholar 

  42. Imbrie J.Z., The ground state of the three-dimensional random field Ising model, Commun. Math. Phys. 98 (1985) 145–176.

    Article  MathSciNet  Google Scholar 

  43. Israel R.B., Existence of phase transitions for long-range interactions, Commun. Math. Phys. 43 (1975) 59–68.

    Article  MathSciNet  MATH  Google Scholar 

  44. Israel, R.B, Convexity in the Theory of Lattice Gases Princeton University Press, Princeton, NJ, 1979.

    MATH  Google Scholar 

  45. Khanin, K.M. and Sinai, Ya. G., Existence of free energy for models with long-range random hamiltonians, J. Stat. Phys. 20 (1979) 573–584.

    Article  MathSciNet  Google Scholar 

  46. Khanin, K. M., Absence of phase transitions in one-dimensional long-range spin systems with random hamiltonian. Theor. Math. Phys. 43 (1980) 445–449.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  48. Laroche, E., Sur des Inégalités de Corrélation et sur les Inégalités de Sobolev Logarithmiques en Mécanique Statistique Thèse Toulouse, 1993.

    Google Scholar 

  49. Ledrappier F., Pressure and variational principle for random Ising model, Commun. Math. Phys. 56 (1977) 297–302.

    Article  MathSciNet  MATH  Google Scholar 

  50. Lu, S.L., Yau, H.-T., Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Commun. Math. Phys. 156 (1993) 399–433.

    Article  MathSciNet  MATH  Google Scholar 

  51. Mezard, M., Parisi, G., and Virasoro, M.A., Spin Glass Theory and Beyond World Scientific, Singapore, 1987.

    MATH  Google Scholar 

  52. Martinelli, F. and Olivieri, E., Approach to equilibrium of Glauber Ddynamics in the one phase region: I. The attractive case, II. The general case, Commun. Math. Phys. 161 (1993) 447–486; 487–514.

    Article  MathSciNet  Google Scholar 

  53. Miekisz, J., Many phases in systems without periodic ground states, Commun. Math. Phys. 107 (1986) 577–586.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  55. Olla, S., Large deviations for Gibbs random fields, Probab. Theor. Rel Fields 77 (1988) 343–357.

    Article  MathSciNet  MATH  Google Scholar 

  56. Park, Y.M., Extension of Pirogov-Sinai theory of phase transitions to infinite range interactions: I. Cluster expansion, II. Phase diagram, Commun. Math. Phys. 114 (1988) 187–216; 219–241.

    Google Scholar 

  57. Picco, P., On the absence of breakdown of symmetry for the plane rotator model with long-range interaction, J. Stat. Phys. 32 (1983) 627–648.

    Article  MathSciNet  Google Scholar 

  58. Picco, P., Upper bound on the decay of correlations in the plane rotator model with long-range random interaction, J. Stat. Phys. 36 (1984) 489.

    Article  MathSciNet  MATH  Google Scholar 

  59. Pirogov, S., Sinai, Ya. G., Phase diagrams of classical spin systems, I, Theor. Math. Phys. 25 (1975) 1185–1192;

    Article  MathSciNet  Google Scholar 

  60. Pirogov, S., Sinai, Ya. G., Phase diagrams of classical spin systems II, Theor. Math. Phys. 26 (1976) 39–49.

    Article  MathSciNet  Google Scholar 

  61. Ruelle, D., Statistical Mechanics: Rigorous Results W.A. Benjamin, Reading, MA, 1969.

    MATH  Google Scholar 

  62. Sherrington, D. and Kirkpatrick, S., Solvable model of a spin-glass, Phys. Rev. Lett. 35 (1975) 1792–1796;

    Article  Google Scholar 

  63. Sherrington, D. and Kirkpatrick, S, Infinite-ranged models of spin-glasses, Phys. Rev. B317 (1978) 4384–4403.

    Google Scholar 

  64. Simon, B., The Statistical Mechanics of Lattice Gases Princeton University Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  65. Sinai, Ya. G., Theory of Phase Transitions: Rigorous Results New York, Pergamon Press, 1982.

    MATH  Google Scholar 

  66. Slawny, J., Low-temperature Properties of Classical Lattice Systems: Phase Transitions and Phase Diagrams in Phase Transitions Domb, C. and Lebowitz, J.L., eds., 11 (1987) 127–205.

    Google Scholar 

  67. Spitzer, F., Phase transition in one-dimensional nearest-neighbor systems, J. Fund. Anal. 20 (1975) 240–255.

    Article  MathSciNet  MATH  Google Scholar 

  68. Stroock, D.W. and Zegarlinski B., The logarithmic Sobolev inequality for continuous spin systems on a lattice, J. Func. Anal. 104 (1992) 299–326.

    Article  MathSciNet  MATH  Google Scholar 

  69. Stroock, D.W. and Zegarlinski B., The equivalence of the logarithmic Sobolev inequality and the Dobrushin–Shlosman mixing condition, Commun. Math. Phys. 144 (1992) 303–323.

    Article  MathSciNet  MATH  Google Scholar 

  70. Stroock, D.W. and Zegarlinski, B., The logarithmic Sobolev inequality for discrete spin systems on a lattice, Commun. Math. Phys. 149 (1992) 175–193.

    Article  MathSciNet  MATH  Google Scholar 

  71. Stroock, D.W. and Zegarlinski, B., The Ergodic Properties of Glauber Dynamics, J. Stat. Phys. 81 (1995) 1007–1019.

    Article  MathSciNet  MATH  Google Scholar 

  72. Vuillermot, P.A., Thermodynamics of quenched random spin systems and application to the problem of phase transition in magnetic-(spin)-glasses, J. Phys. (Math. Gen.) A10 (1977) 1319.

    Article  MathSciNet  Google Scholar 

  73. Zahradnik, M., An Alternate Version of Pirogov-Sinai Theory, Commun. Math. Phys. 93 (1984) 559–581.

    Article  MathSciNet  Google Scholar 

  74. Zahradnik, M., Lecture given at Colloque “Physique mathématique des systèmes désordonnés,” Marseille 27–31 juillet 1992.

    Google Scholar 

  75. Zegarlinski, B., The Gibbs measures and partial differential equations, Commun. Math. Phys. 107 (1986) 411–429.

    Article  MathSciNet  MATH  Google Scholar 

  76. Zegarlinski, B., Spin glasses with long-range interactions at high-temperature, J. Stat. Phys. 47 (1987) 911–930.

    Article  MathSciNet  Google Scholar 

  77. Zegarlinski, B., On equivalence of spin field pictures of lattice systems, J. Stat Phys. 59 (1990) 1511–1530.

    Article  MathSciNet  MATH  Google Scholar 

  78. Zegarlinski, B., Interactions and pressure functional for disordered lattice systems. Commun. Math. Phys. 33 (1991) 305–339.

    Article  MathSciNet  Google Scholar 

  79. Zegarlinski, B., Spin systems with long-range interactions, Rev. Math. Phys. 6 (1992) 115–134.

    Article  MathSciNet  Google Scholar 

  80. Zegarlinski, B., On Log–Sobolev inequalities for infinite lattice systems, Lett. Math. Phys. 20 (1990) 173–182.

    Article  MathSciNet  MATH  Google Scholar 

  81. Zegarlinski, B., Log–Sobolev inequalities for infinite one-dimensional lattice systems, Commun. Math. Phys. 133 (1990) 147–162.

    Article  MathSciNet  MATH  Google Scholar 

  82. Zegarlinski, B., Dobrushin uniqueness theorem and logarithmic Sobolev inequalities, J. Func. Anal 105 (1992) 77–111.

    Article  MathSciNet  MATH  Google Scholar 

  83. Zegarlinski, B., Strong decay to equilibrium in one dimensional random spin systems, J. Stat. Phys. 77 (1994) 717–732.

    Article  MathSciNet  MATH  Google Scholar 

  84. Zygmund, A., Trigonometrical series Vol. 5 Monografje Matematyczne, Warszawa-Lwów, 1935.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Imperial College, London, UK

    Bogusław Zegarlinski

Authors
  1. Bogusław Zegarlinski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Weierstrass — Institut für Angewandte Analysis und Stochastik, D-10117, Berlin, Germany

    Anton Bovier

  2. Centre de Physique Théorique, CNRS-Luminy, 13288, Marseille, France

    Pierre Picco

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Birkhäuser Boston

About this chapter

Cite this chapter

Zegarlinski, B. (1998). Random Spin Systems with Long-Range Interactions. In: Bovier, A., Picco, P. (eds) Mathematical Aspects of Spin Glasses and Neural Networks. Progress in Probability, vol 41. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4102-7_8

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-4102-7_8

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-8653-0

  • Online ISBN: 978-1-4612-4102-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation