Fluctuations for the Ginzburg-Landau {nabla φ} Interface Model on a Bounded Domain
Abstract
We study the massless field on {D_n = D \cap tfrac{1}{n} {Z}^2}, where {D subseteq {R}^2} is a bounded domain with smooth boundary, with Hamiltonian {{H}(h) = sum_{x ∼ y} {V}(h(x) - h(y))}. The interaction {{V}} is assumed to be symmetric and uniformly convex. This is a general model for a (2 + 1)-dimensional effective interface where h represents the height. We take our boundary conditions to be a continuous perturbation of a macroscopic tilt: h( x) = n x · u + f( x) for {x in partial D_n, u in {R}^2}, and f : R 2 → R continuous. We prove that the fluctuations of linear functionals of h( x) about the tilt converge in the limit to a Gaussian free field on D, the standard Gaussian with respect to the weighted Dirichlet inner product {(f,g)_nabla^β = int_D sum_i β_i partial_i f_i partial_i g_i} for some explicit β = β( u). In a subsequent article, we will employ the tools developed here to resolve a conjecture of Sheffield that the zero contour lines of h are asymptotically described by SLE(4), a conformally invariant random curve.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- December 2011
- DOI:
- 10.1007/s00220-011-1315-9
- arXiv:
- arXiv:1002.0381
- Bibcode:
- 2011CMaPh.308..591M
- Keywords:
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- Mathematics - Probability;
- Mathematical Physics;
- 60F05;
- 60G60;
- 60J27
- E-Print:
- 58 pages