LIOUVILLE FIRST-PASSAGE PERCOLATION: SUBSEQUENTIAL SCALING LIMITS AT HIGH TEMPERATURE

• 成果类型: Article
• 署名作者: Ding, Jian; Dunlap, Alexander
• 署名单位: University of Pennsylvania; Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1267
• 发表日期: 2019
• 页码: 690-742
• 关键词: planar random-cluster; MAPS; probabilities; point; MODEL

摘要

Let {Y-B(x) : is an element of B} be a discrete Gaussian free field in a two-dimensional box of B side length S with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex x is given a weight of e(gamma YB(x)) for some gamma > 0. We show that for sufficiently small but fixed gamma > 0, for any sequence of scales {S-k} there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov-Hausdorff sense to a random metric on the unit square in R-2. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Holder-continuous homeomorphisms to the unit square with the Euclidean metric.