Existence of random gradient states

Details Existence of random gradient states Existence of random gradient states

2010-12-20

arxiv.org/abs/1012.4375v3

Mathematics

Probability

31 pages, 1 figure

Scientific paper

We consider two versions of random gradient models. In model A) the interface feels a bulk term of random fields while in model B) the disorder enters though the potential acting on the gradients itself. It is well known that without disorder there are no 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. Van Enter and one of us proved that adding a disorder term as in model A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in $d = 2$. In the present paper we prove the existence of shift-covariant gradient Gibbs measures for model A) when $d\geq 3$ and the expectation with respect to the disorder is zero, and for model B) when $d\geq 2$. When the expectation with respect to the disorder is non-zero in model A), there are no shift-covariant gradient Gibbs measures for $d\ge 3$. We also prove similar results of existence/non-existence of the surface tension for the two models and give the characteristic properties of the respective surface tensions.

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