zbMATH — the first resource for mathematics

Equilibrium fluctuations for \(\nabla\varphi\) interface model. (English) Zbl 1017.60100
The subject of the paper are the space-time fluctuations of the shape of an interface (for instance a phase boundary), which is flat in equilibrium. The deviation from the flat interface (the height) is modeled by a massless scalar field with reversible Langevin dynamics. Neighboring heights are connected through an elastic potential. For a strictly convex elastic potential it is proved that on a large space-time scale these fluctuations are governed by an infinite-dimensional Ornstein-Uhlenbeck process. Its effective diffusion type covariance matrix is characterized through a variational formula.
The paper uses the following techniques: 1. An infinite-dimensional version of an identity of Helffer and Sjöstrand is proved, expressing static correlations as the solution of suitable elliptic partial differential equations. Rewriting them in terms of diffusion processes, a random walk in a dynamic environment is obtained. 2. This representation translates the original fluctuation problem into a problem involving an invariance principle for a certain random walk in a dynamic random environment. The convergence of this random walk to a Brownian motion in \(R^d\) with a specific diffusion matrix is proved. 3. Applying Nash inequality and a generalization of it, it is shown that the dynamic covariance has a limit as \(\varepsilon \to 0.\) 4. Finally, the convergence of the processes is established, using the convergence of the covariance.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
Full Text: DOI
[1] Aronson, D. G. (1967). Bounds on the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 890-896. · Zbl 0153.42002 · doi:10.1090/S0002-9904-1967-11830-5
[2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[3] Brascamp, H. J. and Lieb, E. (1976). On extensions ofthe Brun-Minkowski and Prekopa- Leinler theorems. J. Funct. Anal. 22 366-389. · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5
[4] Brascamp, H. J., Lieb, E. H. and Lebowitz, J. L. (1976). The statistical mechanics ofanharmonic lattices. Bulletin of the International Statistical Institute. Proceedings of the Fortieth Session 1, 393-404. ISI, The Netherlands. · Zbl 0357.60051
[5] Carlen, E., Kusuoka, S. and Stroock, D. (1987). Upper bounds for symmetric Markov transition functions Ann. Inst. H. Poincaré Probab. Statist. 25 245-287. · Zbl 0634.60066 · numdam:AIHPB_1987__23_S2_245_0 · eudml:77309
[6] Chang, C. C. and Yau, H. T. (1992). Fluctuations for one-dimensional Ginzburg-Landau models in nonequilibrium. Comm. Math. Phys. 145 209-234. · Zbl 0754.76006 · doi:10.1007/BF02099137
[7] De Masi, A., Ferrari, P., Goldstein, S. and Wick, D. (1989). An invariance principle for reversible markov processes. Applications to random walk in random environments. J. Statist. Phys. 55 787-855. · Zbl 0713.60041 · doi:10.1007/BF01041608
[8] Deuschel, J.-D., Giacomin, G. and Ioffe, D. (2000). Large deviations and concentration properties for interface models. Probab. Theory Related Fields 117 49-111. · Zbl 0988.82018 · doi:10.1007/s004400000045
[9] Doss, H. and Royer, G. (1978). Processus de diffusion associe aux mesures de Gibbs sur d.Wahrsch. Verw. Gebiete 46 107-124. · Zbl 0384.60076 · doi:10.1007/BF00535690
[10] Fabes, E. and Stroock, D. (1987). The De Giorgi-Moser Harnack principle via the old ideas ofNash. Arch. Rational Mech. Anal. 96 327-338. · Zbl 0652.35052 · doi:10.1007/BF00251802
[11] Fernandez, R., Fr öhlich, J. and Sokal, A. D. (1992). Random Walk, Critical Phenomena, and Triviality in Quantum Field Theory. Springer, New York. · Zbl 0761.60061
[12] Fritz, J. (1982). Infinite lattice systems of interacting diffusion processes, existence and regularity properties.Wahrsch. Gebiete 59 291-309. · Zbl 0458.60096 · doi:10.1007/BF00532222
[13] Funaki, T. and Spohn, H. (1997). Motion by mean curvature from the Ginzburg-Landau interface model. Comm. Math. Phys. 185 1-36. · Zbl 0884.58098 · doi:10.1007/s002200050080
[14] Helffer, B. and Sj östrand, J. (1994). On the correlation for Kac-like models in the convex case. J. Statist. Phys. 74 349-409. · Zbl 0946.35508 · doi:10.1007/BF02186817
[15] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1-19. · Zbl 0588.60058 · doi:10.1007/BF01210789
[16] Kipnis, C. and Landim, C. (1999). Scaling, Limits of Interacting Particles Systems. Springer, New York. · Zbl 0927.60002
[17] Landim, C., Olla, S. and Yau, H. T. (1998). Convection diffusion equations with space-time ergodic random-flow. Probab. Theory Related Fields 112 203-220. · Zbl 0914.60070 · doi:10.1007/s004400050187
[18] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York. · Zbl 0559.60078
[19] Lyons, T. J. and Weian,(1988). A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Astérisque 157-158 249-271. · Zbl 0654.60059
[20] Naddaf, A. and Spencer, T. (1997). On homogenization and scaling limit for some gradient perturbation ofa massless free field. Comm. Math. Phys. 193 55-84. · Zbl 0871.35010 · doi:10.1007/BF02509796
[21] Olla, S. (1994). Homogenization of Diffusion Processes in Random Fields. Lecture notes de École Doctorale. École Polytechnique (Palaiseau).
[22] Shiga, T. and Shimuzo, A. (1980). Infinite-dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20 395-416. · Zbl 0462.60061
[23] Spohn, H. (1991). Large Scale Dynamics of Interacting Particles. Springer, New York. · Zbl 0742.76002
[24] Stroock, D. and Zheng, W. (1997). Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist. 33 619-649. · Zbl 0885.60065 · doi:10.1016/S0246-0203(97)80107-0 · numdam:AIHPB_1997__33_5_619_0 · eudml:77584
[25] Zhu, M. (1990). Equilibrium fluctuations for one-dimensional Ginzburg-Landau lattice model. Nagoya Math. J. 17 63-92. · Zbl 0678.60102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.