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).
Publié le : 1997-07-14
Classification:
Nonquasilocality,
discontinuity of conditional probabilities,
monotonicity preserving specifications,
random fields,
Gibbs measures,
projections of measures,
global Markov property,
decimation processes,
Ising model,
60G60,
60K35,
60J99,
82B20,
82B05,
82B28
@article{1024404514,
author = {Fern\'andez, R. and Pfister, C.-E.},
title = {Global specifications and nonquasilocality of projections of
Gibbs measures},
journal = {Ann. Probab.},
volume = {25},
number = {4},
year = {1997},
pages = { 1284-1315},
language = {en},
url = {http://dml.mathdoc.fr/item/1024404514}
}
Fernández, R.; Pfister, C.-E. Global specifications and nonquasilocality of projections of
Gibbs measures. Ann. Probab., Tome 25 (1997) no. 4, pp. 1284-1315. http://gdmltest.u-ga.fr/item/1024404514/