Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

• 成果类型: Article
• 署名作者: Chiarini, Alberto; Nitzschner, Maximilian
• 署名单位: University of California System; University of California Los Angeles; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00957-7
• 发表日期: 2020
• 页码: 525-575
• 关键词: large deviations; percolation

摘要

We investigate level-set percolation of the discrete Gaussian free field on Zd, d = 3, in the strongly percolative regime. We consider the event that the level-set of theGaussian free field below a level a disconnects the discrete blow-up of a compact set A. Rd from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of theGaussian free field deviate from a specific multiple of the harmonic potential of A, when disconnection occurs. These bounds, combined with the findings of the recent work by Duminil-Copin, Goswami, Rodriguez and Severo, show that conditionally on disconnection, the Gaussian free field experiences an entropic push-down proportional to the harmonic potential of A. In particular, due to the slow decay of correlations, the disconnection event affects the field on the whole lattice. Furthermore, we provide a certain `profile' description for the field in the presence of disconnection. We show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of A, it locally retains the structure of a Gaussian free field shifted by a constant value. Our proofs rely crucially on the `solidification estimates' developed in Nitzschner and Sznitman (to appear in J Eur Math Soc, arXiv:1706.07229).