A Ray-Knight theorem fordelΦ interface models and scaling limits

• 成果类型: Article
• 署名作者: Deuschel, Jean-Dominique; Rodriguez, Pierre-Francois
• 署名单位: Technical University of Berlin; Imperial College London
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01275-3
• 发表日期: 2024
• 页码: 447-499
• 关键词: random interlacements; random-walk; cover times; FIELDS; perturbations; inequalities; Isomorphism; set

摘要

We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent 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 well-known generalization of the second Ray-Knight 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 Wick-ordered square of a Gaussian free field on R-3 with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.