## Harmonic crystal on the wall: A microscopic approach.(English)Zbl 0893.60062

According to physical beliefs the shape of a small droplet touching a wall (under absence of gravitation) and staying in the equilibrium with its vapour can be described in the framework of the Winterbottom construction. The paper contains the first mathematically rigorous result in three dimensions that justifies the corresponding physical prediction for the 2+1 dimensional Gaussian solid-on-solid model. A concentration result (in $$L_1$$ norm) is proven for the limiting shape of the rescaled microscopic Gaussian field in the canonical ensemble under non-negativity condition, which models the presence of the wall, and volume constraint around the deterministic solution of the related macroscopic variational problem of the Winterbottom type. The proof is based on a coarse graining of the random microscopic region “wetted” by the crystal and random walk representations of various quantities related to free massless Gaussian field. Stability properties of the torsional rigidity problem are also investigated.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
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