An isomorphism theorem for GinzburgLandau interface models and scaling limits
Abstract
We introduce a natural measure on biinfinite random walk trajectories evolving in a timedependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which  generically  is not Gaussian. In the quadratic case, we recover a wellknown generalization of the second RayKnight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wickordered square of a Gaussian free field on $\mathbb{R}^3$ with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.14805
 Bibcode:
 2022arXiv220614805D
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 60G60;
 82B20;
 82B41
 EPrint:
 38 pages