Regular g-measures are not always Gibbsian

Roberto Fernandez (University of Utrecht)
Sandro Gallo (Universidade Estadual de Campinas)
Gregory Maillard (CMI-LATP, Aix-Marseille University)

Abstract


Regular g-measures are discrete-time processes determined by conditional expectations with respect to the past. One-dimensional Gibbs measures, on the other hand, are fields determined by simultaneous conditioning on past and future. For the Markovian and exponentially continuous cases both theories are known to be equivalent. Its equivalence for more general cases was an open problem. We present a simple example settling this issue in a negative way: there exist $g$-measures that are continuous and non-null but are not Gibbsian. Our example belongs, in fact, to a well-studied family of processes with rather nice attributes: It is a chain with variable-length memory, characterized by the absence of phase coexistence and the existence of a visible renewal scheme

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Pages: 732-740

Publication Date: November 20, 2011

DOI: 10.1214/ECP.v16-1681

References

  1. R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, 470 (1975) Springer-Verlag, Berlin. Math. Review 442989
  2. R.L. Dobrushin. The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13 (1968) 197-224. Math. Review number not available.
  3. A.C.D. van Enter, R. Fernández, and A.D. Sokal. Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72 (1993) 879-1167. Math. Review 1241537
  4. A.C.D. van Enter and E. Verbitskiy. Erasure entropies and Gibbs measures. Markov Process. Related Fields 16 (2010) 3-14. Math. Review 2664333
  5. R. Fernández. Gibbsianness and non-Gibbsianness in Lattice random fields. In: Mathematical statistical physics (A. Bovier, A.C.D. van Enter, F. den Hollander and F. Dunlop eds.), (2006) 731-799, Elsevier B. V., Amsterdam. Math. Review 2581896
  6. R. Fernández and G. Maillard. Chains with complete connections and one-dimensional Gibbs measures. Electron. J. Probab. 9 (2004) 145-176. Math. Review 2041831
  7. R. Fernández and G. Maillard. Construction of a specification from its singleton part. ALEA 2 (2006) 297-315. Math. Review 2285734
  8. S. Gallo. Chains with unbounded variable length memory: perfect simulation and visible regeneration scheme. Adv. in Appl. Probab. 43 (2011) 735-759. Math. Review number not available.
  9. H.-O. Georgii. Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, 9 (1988) Walter de Gruyter and Co., Berlin. Math. Review 956646
  10. T.E. Harris. On chains of infinite order. Pacific J. Math. 5 (1955) 707-24. Math. Review 75482
  11. S. Kalikow. Random Markov processes and uniform martingales. Isr. J. Math. 71 (1990) 33-54. Math. Review 1074503
  12. M. Keane. Strongly mixing $g$-measures. Inventiones Math. 16 (1972) 309-324. Math. Review 310193
  13. G. Keller. Equilibrium states in ergodic theory. London Mathematical Society Student Texts, 42 (1998) Cambridge University Press, Cambridge. Math. Review 1618769
  14. O.K. Kozlov. Gibbs description of a system of random variables. Probl. Inform. Transmission 10 (1974) 258-265. Math. Review 467970
  15. O.E. Lanford III and D. Ruelle. Observables at infinity and states with short range correlations in statistical mechanics. Comm. Math. Phys. 13 (1969) 194-215. Math. Review 256687
  16. O. Onicescu and G. Mihoc. Sur les chaînes statistiques. C. R. Acad. Sci. Paris 200 (1935) 511-512. Math. Review number not available.
  17. D. Ruelle. Thermodynamic formalism. Encyclopedia of Mathematics 5 (1978) Addison Wesley, New-York. Math. Review 511655
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