Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 13 (2008) > Paper 47 open journal systems 


The Posterior metric and the Goodness of Gibbsianness for transforms of Gibbs measures

Christof Külske, University of Groningen
Alex A Opoku, University of Groningen


Abstract
We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models subjected to local transformations. Such systems arise in the study of a stochastic time-evolution of Gibbs measures or as noisy observations. Assuming no a priori metric on the local state spaces but only a measurable structure, we define the posterior metric on the local image space. We show that it allows in a natural way to divide the local part of the continuity estimates from the spatial part (which is treated by Dobrushin uniqueness here). We show in the concrete example of the time evolution of rotators on the $(q-1)$-dimensional sphere how this method can be used to obtain estimates in terms of the familiar Euclidean metric. In another application we prove the preservation of Gibbsianness for sufficiently fine local coarse-grainings when the Hamiltonian satisfies a Lipschitz property


Full text: PDF

Pages: 1307-1344

Published on: August 25, 2008
Bibliography
  1. H.-O. Georgii. Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics (1988) Math. Review 89k:82010
  2. C. Müller. Spherical Harmonics. Lecture Notes in Mathematics 17 Springer (1966). Math. Review 33:7593
  3. C. Külske and A. Le Ny. Spin-flip dynamics of the Curie-Weiss model: Loss of Gibbsianness with possibly broken symmetry. Commun. Math. Phys. 271, 431-454 (2007). Math. Review 2008f:82058
  4. A.C.D. van Enter, R. Fernàndez, F. den Hollander and F. Redig. Possible Loss and recovery of Gibbsianness during the stochastic evolution of Gibbs Measures. Commun. Math. Phys. 226, 101-130 (2002). Math. Review 2003e:82040
  5. R. Fernàndez. Gibbsianness and non-Gibbsianness in lattice random fields. Les Houches, LXXXIII, (2005). Math. Review number not available.
  6. C. Külske. Concentration Inequalities for Functions of Gibbs Fields with Applications to Diffraction and Random Gibbs Measures. Commun. Math. Phys. 239, 29-51 (2003). Math. Review 2004i:60069
  7. C. Külske, J.-R. Chazottes, P. Collet and F. Redig. Concentration Inequalities for random fields via Coupling. Prob. Theory Relat. Fields 137, 201-225 (2006). Math. Review MR2278456
  8. A.C.D. van Enter and C. Külske. Two connections between random systems and non-Gibbsian measures. J. Stat. Phys. 126, 1007-1024 (2007). Math. Review 2008f:82037
  9. A. Le Ny and F. Redig. Short-time conservation of Gibbsianness under local stochastic evolution. J. Stat. Phys. Volume 109, 1073-1090 (2002). Math. Review 2003i:82060
  10. C. Külske, F. Redig. Loss without recovery of Gibbsianness during diffusion of continuous spins. Prob. Theory Relat. Fields 135, 428-456 (2006). Math. Review 2007g:82036
  11. A.C.D. van Enter, R. Fernàndez and A.D. Sokal. Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory. J. Stat. Phys. 72, 879-1167 (1993). Math. Review 94m:82012
  12. C. Külske. Non-Gibbsianness and phase transitions in random lattice spin models. Markov. Proc. Rel. Fields 5, 357-383 (1999). Math. Review 2001a:82042
  13. T. Lindvall and L.C.G. Rogers. Coupling of multidimensional Diffusion by Reflection. Ann. Probab. 14, 860-872 (1986). Math. Review 88b:60179
  14. 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). Math. Review number not available.
  15. I. Karatzas and S.T. Shreve. Brownian Motion and Stochastic Calculus, 2ed. Springer- Verlag, GTM 113 (1991). Math. Review 92h:60127
  16. P. Lévy. Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris (1948). Math. Review 10,551a
  17. A.C.D. van Enter and W.M. Ruszel. Gibbsianness vs. Non-Gibbsianness of time-evolved planar rotor models. arXiv:0711.3621v1 (2007). Math. Review number not available.
  18. C. Külske and A.A. Opoku. Continuous Spin Mean-Field models: Limiting kernels and Gibbs Properties of local transforms. arXiv:0806.0802 (2008). Math. Review number not available.
  19. R.L. Dobrushin and M. Zahradník. Phase diagrams for continuous-spin models: An extension of the Pirogov-Sinai Theory. Math. Problems of Stat. Phys. and Dynamics, Reidel, 1-123 (1986). Math. Review 88k:82015
  20. R.B. Israel. High-Temperature Analyticity in Classical Lattice Systems. Commun. Math. Phys. 50, 245-257 (1976). Math. Review 56:4578
  21. J.-D. Deuschel. Infinite-dimensional diffusion processes as Gibbs measures on C[0; 1]^(Z^d). Prob. Theory Relat. Fields 76, 325-340 (1987). Math. Review 89m:60252
  22. D. Dereudre and S. Roelly. Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions. J. Stat. Phys. 121, 511-551 (2005). Math. Review 2006m:82094
  23. R. Fernàndez and G. Maillard. Construction of a specification from its singleton part. ALEA Lat. Am. J. Probab. Math. Stat. 2, 297-315 (2006) Math. Review 2007k:60322
  24. P. Auscher, T. Coulhon, X.T. Duong and S. Hofmann. Riesz transform on manifolds and heat kernel regularity. Ann. Sci. Ecole Norm. Sup. 37, 911-957 (2004). Math. Review 2005k:58043
  25. J. Fröhlich, B. Simon and T. Spencer. Infrared bounds, phase transitions and continuous symmetry breaking. Comm. Math. Phys. 50 79-95 (1976). Math. Review 54:9530
  26. P. Bilingsley. Probability and Measure 3rd ed. A Wiley-Interscience Publication (1995). Math. Review 95k:60001
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489