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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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
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