We consider statistical mechanics models of continuous spins in a disordered environment. These models have a natural interpretation as effective interface models. It is well known that without disorder there are no interface Gibbs measures in infinite volume in dimension d=2, while there are “gradient Gibbs measures” describing an infinite-volume distribution for the increments of the field, as was shown by Funaki and Spohn.
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In the present paper we show that adding a disorder term prohibits the existence of such gradient Gibbs measures for general interaction potentials in d=2. This nonexistence result generalizes the simple case of Gaussian fields where it follows from an explicit computation.
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In d=3 where random gradient Gibbs measures are expected to exist, our method provides a lower bound of the order of the inverse of the distance on the decay of correlations of Gibbs expectations w.r.t. the distribution of the random environment.
Publié le : 2008-02-15
Classification:
Random interfaces,
gradient Gibbs measures,
disordered systems,
lower bound on fluctuations,
slow correlation decay,
60K57,
82B24,
82B44
@article{1199890017,
author = {van Enter, Aernout C. D. and K\"ulske, Christof},
title = {Nonexistence of random gradient Gibbs measures in continuous interface models in d=2},
journal = {Ann. Appl. Probab.},
volume = {18},
number = {1},
year = {2008},
pages = { 109-119},
language = {en},
url = {http://dml.mathdoc.fr/item/1199890017}
}
van Enter, Aernout C. D.; Külske, Christof. Nonexistence of random gradient Gibbs measures in continuous interface models in d=2. Ann. Appl. Probab., Tome 18 (2008) no. 1, pp. 109-119. http://gdmltest.u-ga.fr/item/1199890017/