The Annals of Applied Probability

Existence of random gradient states

Codina Cotar and Christof Külske

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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 through the potential acting on the gradients. It is well known that for gradient models 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 gradients of the field, as was shown by Funaki and Spohn. Van Enter and Külske 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 with a given tilt $u\in \mathbb{R}^{d}$ for model A when $d\geq3$ and the disorder has mean zero, and for model B when $d\geq1$. When the disorder has nonzero mean in model A, there are no shift-covariant gradient Gibbs measures for $d\ge3$. We also prove similar results of existence/nonexistence of the surface tension for the two models and give the characteristic properties of the respective surface tensions.

Article information

Ann. Appl. Probab., Volume 22, Number 4 (2012), 1650-1692.

First available in Project Euclid: 10 August 2012

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Zentralblatt MATH identifier

Primary: 60K57 82B24: Interface problems; diffusion-limited aggregation 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Random interfaces gradient Gibbs measures disordered systems Green’s function surface tension


Cotar, Codina; Külske, Christof. Existence of random gradient states. Ann. Appl. Probab. 22 (2012), no. 4, 1650--1692. doi:10.1214/11-AAP808.

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