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Fluctuations for the Ginzburg-Landau $$\nabla \phi$$ interface model on a bounded domain. (English) Zbl 1237.82030
A general model is considered for a (2+1)-dimensional effective (Ginzburg-Landau) interface. The main result of this paper is the central limit theorem for linear functionals of the height of the interface, defined in terms of a massless field on a subgraph induced in a bounded domain with a smooth boundary. The covariance matrix is explicitly given in the proof, which is a novelty for gradient Gibbs states. The boundary conditions are considered to be a continuous perturbation of a macroscopic tilt. It is proven that fluctuations of linear functionals of the interface height about the tilt converge to a Gaussian free field. Major techniques are developed as a preperatory step to resolve the conjecture due to S. Sheffield [Random surfaces. Astérisque 304. Paris: Société Mathématique de France (SMF) (2005; Zbl 1104.60002)] that the zero contour lines of the height are asymptotically described by a conformally invariant random curve. This issue is postponed to a subsequent article.

##### MSC:
 82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) 35L70 Second-order nonlinear hyperbolic equations 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics 37H10 Generation, random and stochastic difference and differential equations 60F05 Central limit and other weak theorems 60K37 Processes in random environments
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