The Annals of Probability

Global specifications and nonquasilocality of projections of Gibbs measures

R. Fernández and C.-E. Pfister

Full-text: Open access


We study the question of whether the quasilocality (continuity, almost Markovianness) property of Gibbs measures remains valid under a projection on a sub-$\sigma$-algebra. Our method is based on the construction of global specifications, whose projections yield local specifications for the projected measures. For Gibbs measures compatible with monotonicity preserving local specifications, we show that the set of configurations where quasilocality is lost is an event of the tail field. This set is shown to be empty whenever a strong uniqueness property is satisfied, and of measure zero when the original specification admits a single Gibbs measure. Moreover, we provide a criterion for nonquasilocality (based on a quantity related to the surface tension). We apply these results to projections of the extremal measures of the Ising model. In particular, our nonquasilocality criterion allows us to extend and make more complete previous studies of projections to a sublattice of one less dimension (Schonmann example).

Article information

Ann. Probab. Volume 25, Number 3 (1997), 1284-1315.

First available in Project Euclid: 18 June 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G60: Random fields 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J99: None of the above, but in this section
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B05: Classical equilibrium statistical mechanics (general) 82B28: Renormalization group methods [See also 81T17]

Nonquasilocality discontinuity of conditional probabilities monotonicity preserving specifications random fields Gibbs measures projections of measures global Markov property decimation processes Ising model


Fernández, R.; Pfister, C.-E. Global specifications and nonquasilocality of projections of Gibbs measures. Ann. Probab. 25 (1997), no. 3, 1284--1315. doi:10.1214/aop/1024404514.

Export citation


  • Albeverio, S. and Zegarlinski, B. (1992). Global Markov property in quantum field theory and statistical mechanics. In Ideas and Methods in Quantum and Statistical Physics (S. Albeverio, J. E. Fenstad, H. Holden and T. Lindstrøm, eds.) 331-369. Cambridge Univ. Press.
  • Dobrushin, R. L. (1968). Gibbsian random fields for lattice systems with pairwise interactions. Functional Anal. Appl. 3 22-28.
  • F ¨ollmer, H. (1980). On the global Markov property. In Quantum Fields: Algebras, Processes (L. Streit, ed.) 293-302. Springer, New York.
  • Fr ¨ohlich, J. and Pfister, C.-E. (1987). Semi-infinite Ising model II: the wetting and layering transitions. Comm. Math. Phys. 112 51-74.
  • Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
  • Goldstein, S. (1978). A note on specifications. Z. Wahrsch. Verw. Gebiete 46 45-51.
  • Goldstein, S. (1980). Remarks on the global Markov property. Comm. Math. Phys. 74 223-234.
  • Griffiths, R. B. and Pearce, P. A. (1979). Mathematical properties of renormalization-group transformations. J. Statist. Phys. 20 499-545.
  • Grimmett, G. (1995). The stochastic random-cluster process, and the uniqueness of randomcluster measures. Ann. Probab. 23 1461-1510.
  • Israel, R. B. (1981). Banach algebras and Kadanoff transformations. In Random FieldsRigorous Results in Statistical Mechanics and Quantum Field Theory Vol. II. Coll. Math. Soc. Janos Bolyai 27 593-608. North-Holland, Amsterdam.
  • Kozlov, O. K. (1974). Gibbs description of a system of random variables. Problems Inform. Transmission 10 258-265.
  • Lanford, O. E. and Ruelle, D. (1969). Observables at infinity and states with short range correlations in statistical mechanics. Comm. Math. Phys. 13 194-215.
  • Lebowitz, J. L. and Pfister, C.-E. (1981). Surface tension and phase coexistence. Phys. Rev. Lett. 46 1031-1033.
  • L ¨orinczi, J. (1994). Some results on the projected two-dimensional Ising model. In On Three Levels (M. Fannes, C. Maes and A. Verbeure, eds.) 373-380. Plenum, New York. L ¨orinczi, J. (1995a). On limits of the Gibbsian formalism in thermodynamics. Ph.D. dissertation, Univ. Groningen. L ¨orinczi, J. (1995b). Quasilocality of projected Gibbs measures through analyticity techniques. Helv. Phys. Acta 68 605-626.
  • L ¨orinczi, J. and Vande Velde, K. (1994). A note on the projection of Gibbs measures. J. Statist. Phys. 77 881-887.
  • Maes, C. and Vande Velde, K. (1992). Defining relative energies for the projected Ising measure. Helv. Phys. Acta 65 1055-1068.
  • Martinelli, F. and Olivieri, E. (1993). Some remarks on pathologies of the renormalizationgroup transformations for the Ising model. J. Statist. Phys. 72 1169-1177.
  • Sokal, A. D. (1981). Existence of compatible families of proper regular conditional probabilities. Z. Wahrsch. Verw. Gebiete 56 537-548.
  • Schonmann, R. H. (1989). Projections of Gibbs measures may be non-Gibbsian. Comm. Math. Phys. 124 1-7.
  • Sullivan, W. G. (1973). Potentials for almost Markovian random fields. Comm. Math. Phys. 33 61-74.
  • van Enter, A. C. D., Fern´andez, R. and Sokal, A. D. (1993). Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Statist. Phys. 72 879-1167.
  • van Enter, A. C. D. and L ¨orinczi, J. (1996). Robustness of non-Gibbsian property: some examples. J. Phys. A 29 2465-2473.