Existence and uniqueness of the Liouville quantum gravity metric for γ ∈(0,2)

• 成果类型: Article
• 署名作者: Gwynne, Ewain; Miller, Jason
• 署名单位: University of Cambridge
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00991-6
• 发表日期: 2021
• 页码: 213-333
• 关键词: GAUSSIAN MULTIPLICATIVE CHAOS; planar maps; 1st-passage percolation; fractal structure; SCALING LIMITS; heat kernel; geometry; geodesics; sle; reversibility

摘要

We show that for each gamma is an element of (0, 2), there is a unique metric (i.e., distance function) associated with gamma-Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) h, there is a unique random metric D-h associated with the Riemannian metric tensor e(gamma h)(dx(2) + dy(2)) on C which is characterized by a certain list of axioms: it is locally determined by h and it transforms appropriately when either adding a continuous function to h or applying a conformal automorphism of C (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The gamma-LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding et al. (Tightness of Liouville first passage percolation for gamma is an element of(0, 2), 2019. arXiv:1904.08021) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when gamma = root 8/3, our metric coincides with the root 8/3-LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general gamma is an element of (0, 2), we conjecture that our metric is the Gromov-Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems.