Existence and uniqueness of the Liouville quantum gravity metric for $$\gamma \in (0,2)$$

Abstract

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.

The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

“…The following theorem was proven in [DDDF19,GM19d,DFG+19,GM19a,GM19c,GM19b]. Theorem 1.5 (Existence and uniqueness of the LQG metric).…”

“…There is also an earlier construction of the LQG metric in the special case when γ = √ 8/3 due to Miller and Sheffield [MS15,MS16a,MS16b]. This construction uses a completely different regularization procedure from the one in [DDDF19,GM19c] which only works for γ = √ 8/3. However, the Miller-Sheffield construction gives additional information about the √ 8/3-LQG metric which is not apparent from the construction in [DDDF19,GM19c], such as certain Markov properties for LQG metric balls and the connection to the Brownian map [Le 13,Mie13].…”

“…This metric will be our primary interest in this paper. This γ -LQG metric for general γ ∈ (0, 2) was constructed in [DDDF19,GM19c]. More precisely, [DDDF19] established the tightness of the approximating metrics and [GM19c] showed that the limiting metric is unique and scale invariant in the appropriate sense, building on results from [GM19d,DFG+19,GM19a].…”

“…This γ -LQG metric for general γ ∈ (0, 2) was constructed in [DDDF19,GM19c]. More precisely, [DDDF19] established the tightness of the approximating metrics and [GM19c] showed that the limiting metric is unique and scale invariant in the appropriate sense, building on results from [GM19d,DFG+19,GM19a]. See also [DD19,DF18,DD18] for earlier tightness results (preceding [DDDF19]); and [GM19b,GP19b] which establish the conformal coordinate change formula and the KPZ formula [KPZ88,DS11], respectively, for the LQG metric.…”

Uniqueness of the Welding Problem for Sle and Liouville Quantum Gravity

“…We also remark that the metric (i.e., distance function) for LQG was first constructed in the case that γ = √ 8/3 in [35,39,40] and recently for all γ ∈ (0, 2) in [7,9,[17][18][19][20]. The study of LQG surfaces is motivated in part because they have been conjectured to describe the scaling limits of random planar maps decorated by an instance of a statistical physics model.…”