We show that for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , there is a unique metric (i.e., distance function) associated with $$\gamma $$ γ -Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) h , there is a unique random metric $$D_h$$ D h associated with the Riemannian metric tensor “ $$e^{\gamma h} (dx^2 + dy^2)$$ e γ h ( d x 2 + d y 2 ) ” on $${\mathbb {C}}$$ 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 $$\mathbb {C}$$ C (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The $$\gamma $$ γ -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 $$\gamma \in (0,2)$$ γ ∈ ( 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 $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , our metric coincides with the $$\sqrt{8/3}$$ 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 $$\gamma \in (0,2)$$ γ ∈ ( 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.

中文翻译：

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$$\gamma \in (0,2)$$ 的 Liouville 量子引力度量的存在性和唯一性

我们表明，对于每个 $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) ，都有一个与 $$\gamma $$ γ -Liouville 量子引力相关的唯一度量（即距离函数） (LQG)。更准确地说，我们表明对于全平面高斯自由场 (GFF) h ，存在与黎曼度量张量“ $$e^{\gamma h} (dx ^2 + dy^2)$$ e γ h ( dx 2 + dy 2 ) ”在 $${\mathbb {C}}$$ C 上，其特征在于某个公理列表：它由 h 和当向 h 添加连续函数或应用 $$\mathbb {C}$$ C 的共形自同构（即，复杂的仿射变换）时，它会适当地转换。可以使用局部绝对连续性来构建与 GFF 的其他变体相关联的度量。$$\gamma $$ γ -LQG 度量可以明确构建为 Liouville 第一通道渗透 (LFPP) 的缩放限制，这是通过对 GFF 的缓和版本取幂获得的随机度量。丁等人的早期工作。（Liouville first pass percolation for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , 2019. arXiv:1904.08021 ）表明 LFPP 承认非平凡的后续限制。这篇论文表明后继极限是唯一的并且满足我们的公理列表。在 $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 的情况下，我们的度量与之前工作中构建的 $$\sqrt{8/3}$$ 8 / 3 -LQG 度量重合米勒和谢菲尔德，这又相当于 GFF 的某个变体的布朗图。对于一般的 $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) ，我们推测我们的度量是适当加权随机平面地图模型的 Gromov-Hausdorff 极限，配备了它们的图距离。我们列出了大量未解决的问题。