Codina Cotar and Christof Külske
Source: Ann. Appl. Probab. Volume 22, Number 4 (2012), 1650-1692.

Abstract

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.

First Page:
Primary Subjects: 60K57, 82B24, 82B44
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1344614207
Digital Object Identifier: doi:10.1214/11-AAP808
Zentralblatt MATH identifier: 06083947
Mathematical Reviews number (MathSciNet): MR2985173

References

[1] Akcoglu, M. A. and Krengel, U. (1981). Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323 53–67.
Mathematical Reviews (MathSciNet): MR611442
[2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
Mathematical Reviews (MathSciNet): MR233396
[3] Biskup, M. and Kotecký, R. (2007). Phase coexistence of gradient Gibbs states. Probab. Theory Related Fields 139 1–39.
Mathematical Reviews (MathSciNet): MR2322690
Zentralblatt MATH: 1120.82003
Digital Object Identifier: doi:10.1007/s00440-006-0013-6
[4] Bovier, A. and Külske, C. (1994). A rigorous renormalization group method for interfaces in random media. Rev. Math. Phys. 6 413–496.
Mathematical Reviews (MathSciNet): MR1305590
Digital Object Identifier: doi:10.1142/S0129055X94000171
[5] Bovier, A. and Külske, C. (1996). There are no nice interfaces in $(2+1)$-dimensional SOS models in random media. J. Stat. Phys. 83 751–759.
Mathematical Reviews (MathSciNet): MR1386357
Zentralblatt MATH: 1081.82571
Digital Object Identifier: doi:10.1007/BF02183747
[6] Bricmont, J., El Mellouki, A. and Fröhlich, J. (1986). Random surfaces in statistical mechanics: Roughening, rounding, wetting,.... J. Stat. Phys. 42 743–798.
Mathematical Reviews (MathSciNet): MR833220
Digital Object Identifier: doi:10.1007/BF01010444
[7] Brydges, D. and Yau, H.-T. (1990). Grad $\varphi$ perturbations of massless Gaussian fields. Comm. Math. Phys. 129 351–392.
Mathematical Reviews (MathSciNet): MR1048698
Zentralblatt MATH: 0705.60101
Digital Object Identifier: doi:10.1007/BF02096987
Project Euclid: euclid.cmp/1104180749
[8] Cotar, C. and Deuschel, J. D. (2012). Decay of covariances, uniqueness of ergodic component and scaling limit for a class of $\nabla \varphi$ systems with nonconvex potential. Ann. Inst. H. Poincaré Probab. Statist. 819–853.
[9] Cotar, C., Deuschel, J.-D. and Müller, S. (2009). Strict convexity of the free energy for a class of non-convex gradient models. Comm. Math. Phys. 286 359–376.
Mathematical Reviews (MathSciNet): MR2470934
Zentralblatt MATH: 1173.82010
Digital Object Identifier: doi:10.1007/s00220-008-0659-2
[10] Cotar, C. and Külske, C. Uniqueness of random gradient states. Unpublished manuscript.
[11] den Hollander, F. (2009). Random Polymers. Lecture Notes in Math. 1974. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR2504175
[12] Deuschel, J.-D., Giacomin, G. and Ioffe, D. (2000). Large deviations and concentration properties for $\nabla\varphi$ interface models. Probab. Theory Related Fields 117 49–111.
Mathematical Reviews (MathSciNet): MR1759509
Zentralblatt MATH: 0988.82018
Digital Object Identifier: doi:10.1007/s004400050266
[13] Fröhlich, J. and Pfister, C. (1981). On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems. Comm. Math. Phys. 81 277–298.
Mathematical Reviews (MathSciNet): MR632763
Digital Object Identifier: doi:10.1007/BF01208901
Project Euclid: euclid.cmp/1103920246
[14] Funaki, T. (2005). Stochastic interface models. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1869 103–274. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR2228384
Zentralblatt MATH: 1119.60081
Digital Object Identifier: doi:10.1007/11429579_2
[15] Funaki, T. (2006). The Brascamp-Lieb inequality and its applications. Available at http://www.ms.u-tokyo.ac.jp/~funaki/publ/Pisa06.pdf.
[16] Funaki, T. and Spohn, H. (1997). Motion by mean curvature from the Ginzburg–Landau $\nabla\varphi$ interface model. Comm. Math. Phys. 185 1–36.
Mathematical Reviews (MathSciNet): MR1463032
Zentralblatt MATH: 0884.58098
Digital Object Identifier: doi:10.1007/s002200050080
[17] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
Mathematical Reviews (MathSciNet): MR956646
[18] Giacomin, G., Olla, S. and Spohn, H. (2001). Equilibrium fluctuations for $\nabla \varphi$ interface model. Ann. Probab. 29 1138–1172.
Mathematical Reviews (MathSciNet): MR1872740
Zentralblatt MATH: 1017.60100
Digital Object Identifier: doi:10.1214/aop/1015345767
Project Euclid: euclid.aop/1015345600
[19] Kallenberg, O. (1984). Random Measures. Akademie-Verlag, Berlin.
[20] Komlós, J. (1967). A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18 217–229.
Mathematical Reviews (MathSciNet): MR210177
Zentralblatt MATH: 0228.60012
Digital Object Identifier: doi:10.1007/BF02020976
[21] Külske, C. and Orlandi, E. (2006). A simple fluctuation lower bound for a disordered massless random continuous spin model in $D=2$. Electron. Commun. Probab. 11 200–205 (electronic).
Mathematical Reviews (MathSciNet): MR2266710
Zentralblatt MATH: 1119.82019
[22] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
Mathematical Reviews (MathSciNet): MR1117680
[23] Lawler, G. F., Bramson, M. and Griffeath, D. (1992). Internal diffusion limited aggregation. Ann. Probab. 20 2117–2140.
Mathematical Reviews (MathSciNet): MR1188055
Digital Object Identifier: doi:10.1214/aop/1176989542
Project Euclid: euclid.aop/1176989542
[24] Liggett, T. M. (1985). An improved subadditive ergodic theorem. Ann. Probab. 13 1279–1285.
Mathematical Reviews (MathSciNet): MR806224
Zentralblatt MATH: 0579.60023
Digital Object Identifier: doi:10.1214/aop/1176992811
Project Euclid: euclid.aop/1176992811
[25] Messager, A., Miracle-Solé, S. and Ruiz, J. (1992). Convexity properties of the surface tension and equilibrium crystals. J. Statist. Phys. 67 449–470.
Mathematical Reviews (MathSciNet): MR1171142
Zentralblatt MATH: 0900.82029
Digital Object Identifier: doi:10.1007/BF01049716
[26] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
Mathematical Reviews (MathSciNet): MR900810
[27] Schürger, K. (1988). Almost subadditive multiparameter ergodic theorems. Stochastic Process. Appl. 29 171–193.
Mathematical Reviews (MathSciNet): MR958498
Zentralblatt MATH: 0664.60042
Digital Object Identifier: doi:10.1016/0304-4149(88)90036-1
[28] Simon, B. (1974). The $P(\varphi)_{2}$ Euclidean (quantum) Field Theory. Princeton Univ. Press, Princeton, NJ.
Mathematical Reviews (MathSciNet): MR489552
Zentralblatt MATH: 1175.81146
[29] van Enter, A. C. D. and Külske, C. (2008). Nonexistence of random gradient Gibbs measures in continuous interface models in $d=2$. Ann. Appl. Probab. 18 109–119.
Mathematical Reviews (MathSciNet): MR2380893
Zentralblatt MATH: 1141.60074
Digital Object Identifier: doi:10.1214/07-AAP446
Project Euclid: euclid.aoap/1199890017
[30] van Enter, A. C. D. and Shlosman, S. (2002). First-order transitions for $n$ vector models in two and more dimensions: Rigorous proof. Phys. Rev. Lett. 89 1–3.
[31] van Enter, A. C. D. and Shlosman, S. B. (2005). Provable first-order transitions for nonlinear vector and gauge models with continuous symmetries. Comm. Math. Phys. 255 21–32.
Mathematical Reviews (MathSciNet): MR2123375
Digital Object Identifier: doi:10.1007/s00220-004-1286-1