Fluctuations for the GinzburgLandau {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/s0022001113159
 arXiv:
 arXiv:1002.0381
 Bibcode:
 2011CMaPh.308..591M
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 60F05;
 60G60;
 60J27
 EPrint:
 58 pages