The Annals of Probability
- Ann. Probab.
- Volume 25, Number 3 (1997), 1284-1315.
Global specifications and nonquasilocality of projections of Gibbs measures
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).
Ann. Probab. Volume 25, Number 3 (1997), 1284-1315.
First available in Project Euclid: 18 June 2002
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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. https://projecteuclid.org/euclid.aop/1024404514