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Thermodynamic and scaling limits of the non-Gaussian membrane model. (English) Zbl 1512.82016

Summary: We characterize the behavior of a random discrete interface \(\phi\) on \({[-L,L]^d}\cap{\mathbb{Z}^d}\) with energy \(\sum V(\Delta \phi (x))\) as \(L\to \infty\), where \(\Delta\) is the discrete Laplacian and \(V\) is a uniformly convex, symmetric, and smooth potential. The interface \(\phi\) is called the non-Gaussian membrane model. By analyzing the Helffer-Sjöstrand representation, associated to \(\Delta \phi\), we provide a unified approach to continuous scaling limits of the rescaled and interpolated interface in dimensions \(d=2\), \(3\), Gaussian approximation in negative regularity spaces for all \(d\ge 2\), and the infinite volume limit in \(d\ge 5\).

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60F05 Central limit and other weak theorems
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
35Q82 PDEs in connection with statistical mechanics

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