The Annals of Probability

Global specifications and nonquasilocality of projections of Gibbs measures

R. Fernández and C.-E. Pfister

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Abstract

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

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

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404514

Digital Object Identifier
doi:10.1214/aop/1024404514

Mathematical Reviews number (MathSciNet)
MR1457620

Zentralblatt MATH identifier
0895.60096

Subjects
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]

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

Citation

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


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