Two connections between random systems and non-Gibbsian measures

Journal of Statistical Physics

Volumn 126, Issue 4-5, 2007, Pages 1007-1024

  Van Enter, Aernout C D ^a   Külske, Christof ^a  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

Author keywords

Kac limits; Non Gibbsian measures; Quenched disorder

Indexed keywords

EID: 34047201918     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-006-9185-9     Document Type: Conference Paper

References (55)

1. L. Bertini, E. M. N. Cirillo and E. Olivieri, Gibbsian properties and convergence of the iterates for the block-averaging transformation. Mark. Proc. Rel. Fields 10:381-394 (2004).
2. L. Bertini, E. M. N. Cirillo and E. Olivieri, Graded cluster expansion for lattice systems. Comm. Math. Phys. 258:405-443 (2005).
3. L. Bertini, E. M. N. Cirillo and E. Olivieri, Renormalization group in the uniqueness region Weak Gibbsianity and convergence. Comm. Math. Phys. 261:323-378 (2006).
4. A. Bovier and C. Külske, Coarse-graining techniques for (random) Kac models, Interacting Stochastic Systems Springer Berlin. (2005), pp. 11-28.
5. A. Bovier and M. Zahradnik, The low-temperature phase of Kac-Ising models. J. Stat. Phys. 87:311-332 (1997).
6. A. Bovier and M. Zahradník, Cluster expansions and Pirogov-Sinai theory for long range spin systems. Markov Proc. Rel. Fields 8:443-478 (2002).
7. M. Cassandro, R. Marra and E. Presutti, Corrections to the critical temperature in 2D Ising systems with Kac potentials. J. Stat. Phys. 78:1131-1138(1995).
8. M. Cassandro, E. Orlandi and P. Picco, Typical configurations for one-dimensional random field Kac model. Ann. Prob. 27:1414-1467 (1999).
9. M. Cassandro, E. Orlandi and E. Presutti, Interfaces and typical Gibbs configurations for one-dimensional Kac potentials. Prob. Theor. Rel. Fields 96:57-96 (1993).
10. J. Bricmont and A. Kupiainen, Phase transition in the 3D random field Ising-model. Comm. Math. Phys. 116:539-572(1988).
11. J. Bricmont, A. Kupiainen and R. Lefevere, Renormalisation group pathologies and the definition of Gibbs states. Comm. Math. Phys. 194:359-388 (1998).
12. J. Bricmont, A. Kupiainen and R. Lefevere, Renormalizing the renormalization group pathologies. Phys. Rep. 348:5-31 (2001).
13. T. W. Burkhardt and J. M. J. van Leeuwen, Progress and problems in real-space renormalization. In vol. 30, Topics in Current Physics, Real-Space renormalization, T. W. Burkhardt and J. M. J. van Leeuwen, (eds.), pp. 1-31 Springer, Berlin (1982).
14. D. Dereudre and S. Roelly, Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions. J. Stat. Phys. 121:511-551 (2005).
15. A. C. D. van Enter, R. Fernández and A. D. Sokal, Regularity properties of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory. J. Stat. Phys. 72:879-1167 (1993).
16. A. C. D. van Enter, R. Fernández, F. den Hollander and F. Redig, Possible loss and recovery of gibbsianness during the stochastic evolution of gibbs measures. Comm. Math. Phys. 226:101-130 (2002).
17. A. C. D. van Enter and J. Lörinczi, Robustness of the non-Gibbsian property: Some examples. J. Phys. A, Math. Gen. 29:2465-2473 (1996).
18. A. C. D. van Enter, C. Külske and C. Maes, Comment on: Critical behavior of the randomly spin diluted 2D Ising model: A grand ensemble aproach, by R. Kühn. Phys. Rev. Lett. 84:6134 (2000).
19. A. C. D. van Enter, C. Maes, R. H. Schonmann and S. B. Shlosman, The Griffiths singularity random field. In R. A. Minlos, S. B. Shlosman and Yu. M. Suhov (eds.) On Dobrushin 's Way: From Probability Theory to Statistical Physics pp. 51-58, AMS Translations: Series 2 (Providence, RI, USA, 2000)
20. A. C. D. van Enter, K. Netočný and H. G. Schaap, On the Ising model with random boundary condition. J. Stat. Phys. 118:997-1056 (2005).
21. A. C. D. van Enter and E. A. Verbitskiy, On the variational principle for Generalized Gibbs Measures. Mark. Proc. Rel. Fields 10:411-434 (2004).
22. R. Fernández, Gibbsianness and non-Gibbsianness in lattice random fields. In A. Bovier, J. Dalibard, F. Dunlop, A. C. D. van Enter and F. den Hollander, (eds.), Proceedings 2005 les Hauches Summerschool on Mathematical Statistical Physics, to appear, (2006).
23. J. Fröhlich and J. Z. Imbrie, Improved perturbation expansion for disordered systems: Beating Griffiths singularities. Comm. Math. Phys. 96:145-180 (1984).
24. H.-O. Georgii, Gibbs Measures and Phase Transitions. (de Gruyter, Berlin 1988).
25. T. Gobron and I. Merola, Phase transition induced by increasing the range of interaction in Potts Model, mp-arc 05-272.
26. R. B. Griffiths, Nonanalytic behavior above critical point in a random ising ferromagnet, Phys. Rev. Lett. 23:17-19(1969).
27. O. Häggström and C. Külske, Gibbs properties of the fuzzy Potts model on trees and in mean field. Markov Proc. Rel. Fields 10:477-506 (2004).
28. R. B. Israel, Some generic results in mathematical physics. Mark. Proc. Rel. Fields 10:517-521 (2004).
29. M. Kac, G. Uhlenbeck and P. C. Hemmer, On the van der Waals theory of vapour-liquid equilibrium. I. Discussion of a one-dimensional model. J. Math. Phys. 4:216-228 (1963);
30. II. Discussion of the distribution functions. J. Math. Phys. 4:229-247 (1963);
31. III. Discussion of the critical region. J. Math. Phys. 5:60-74(1964).
32. T. Kennedy, Majority Rule at Low Temperatures for the Square Lattice with 6 = 2 and for the Triangular lattice. J. Star. Phys. 86:1089-1107 (1997).
33. R. Kühn, Equilibrium ensemble approach to disordered systems. 1. General theory and exact results. Z. Phys. B 100:231-242 (1996).
34. R. Kühn, Gibbs versus Non-Gibbs in the equilibrium approach to disordered systems. Mark. Proc. Rel. Fields 10:523-545 (2004).
35. R. Kühn and G. Mazzeo, Critical behaviour of the 2-d spin diluted ising model via the equilibrium ensemble approach. Phys. Rev. E 60:3823-3836 (1999).
36. C. Külske, (Non-)Gibbsianness and phase transitions in random lattice spin models. Mark. Proc. Rel. Fields 5:357-383 (1999).
37. C. Külske, On the Gibbsian nature of the random field Kac model after block-averaging. J. Stat. Phys. 104:991-1012(2001).
38. C. Külske, Weakly Gibbsian Representations for joint measures of quenched lattice spin models. Prob. Theor. Rel. Fields 119:1-30 (2001).
39. C. Külske, Analogues of non-Gibbsianness in joint measures of mean field models. J. Stat. Phys. 112:1101-1130(2003).
40. C. Külske, How non-Gibbsianness helps a metastable Morita minimizer to provide a stable free energy. Mark. Proc. Rel. Fields 10:547-564 (2004).
41. C. Külske and A. Le Ny, Spin-flip dynamics of the Curie-Weiss model: Loss of Gibbsianness with possibly broken symmetry. Preprint, mp-arc 05-357, 2005.
42. C. Külske, A. Le Ny and F. Redig, Relative entropy and variational properties of generalized Gibbsian measures. Ann. Prob. 32:1691-1726 (2004).
43. C. Külske and F. Redig, Loss without recovery of Gibbsianness during diffusion of continuous spins. Prob. Theor. Rel. Fields 135(3): 428-456 (2006).
44. J. L. Lebowitz and C. Maes, The effect of an external field on an interface, entropic repulsion. J. Stat. Phys. 46:39-49 (1987).
45. J. L. Lebowitz and R. H. Schonmann, Pseudo-free energies and large deviations for non gibbsian FKG measures. Prob. Theor. Rel. Fields 77:49-64 (1988).
46. J. Lebowitz and O. Penrose, Rigorous treatment of the Van der Waals Maxwell theory of the liquid-vapour transition. J. Math. Phys. 7:98-113 (1966).
47. R. Lefevere and H. Tasaki, High-temperature expansion for nonequilibrium steady states in driven lattice gases. Phys. Rev. Lett. 94:200601 (2005).
48. C. Maes, A. van Moffaert and F. Redig. Almost Gibbsian versus weakly Gibbsian measures. Stoch. Proc. Appl. 79:1-15(1999).
49. T. Morita, Statistical mechanics of quenched solid solutions with application to magneticallly dilute alloys. J. Math. Phys. 5:1402-1405 (1964).
50. Mark. Proc. Rel. Fields, 10 Proceedings Gibbs-non-Gibbs, Eindhoven. (2004).
51. C. M. Newman, Topics in Disordered Systems, Lecture in Mathematics, ETH-Zürich. Birkhäuser, Basel, (1997).
52. E. Presutti, Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics. Book in preparation, 2006.
53. R. H. Schonmann, Projections of Gibbs measures may be non-Gibbsian. Comm. Math. Phys. 124:1-7 (1989).
54. M. Zahradník, On the structure of low-temperature phases in three-dimensional spin models with random impurities: A general Pirogov-Sinai approach. In R. Kotecký (ed.), Phase Transitions: Mathematics, Physics, Biology, Singapore, World Scientfic (1992), pp. 225-237.
55. M. Zahradnik, Cluster expansions of small contours in abstract Pirogov-Sinai models. Markov Process. Related Fields 8:383-441 (2002).