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.
We prove that for each γ ∈ (0, 2), there is an exponent d γ > 2, the "fractal dimension of γ-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the γ-LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of γ-LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that d γ is a continuous, strictly increasing function of γ and prove upper and lower bounds for d γ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for γ = √ 2 (which corresponds to spanning-tree weighted planar maps) our bounds give 3.4641 ≤ d √ 2 ≤ 3.63299 and in the limiting case we get 4.77485 ≤ lim γ→2 − d γ ≤ 4.89898.
We introduce a general technique for proving estimates for certain random planar maps which belong to the γ-Liouville quantum gravity (LQG) universality class for γ ∈ (0, 2). The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; γ = 8/3); and planar maps weighted by the number of spanning trees (γ = √ 2), bipolar orientations (γ = 4/3) or Schnyder woods (γ = 1) they admit.Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (1993) prediction for the Hausdorff dimension of γ-LQG and we establish the existence of an exponent for certain distances in the map.The basic idea of our approach is to compare a given random planar map M to a mated-CRT map-a random planar map constructed from a correlated two-dimensional Brownian motion-using a strong coupling (Zaitsev, 1998) of the encoding walk for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when γ = 8/3, we instead deduce estimates for the 8/3-mated-CRT map from known results for the UIPT.The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.
We prove that the uniform infinite half-plane quadrangulation (UIHPQ), with either general or simple boundary, equipped with its graph distance, its natural area measure, and the curve which traces its boundary, converges in the scaling limit to the Brownian half-plane. The topology of convergence is given by the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) metric on curve-decorated metric measure spaces, which is a generalization of the Gromov-Hausdorff metric whereby two such spaces (X1, d1, µ1, η1) and (X2, d2, µ2, η2) are close if they can be isometrically embedded into a common metric space in such a way that the spaces X1 and X2 are close in the Hausdorff distance, the measures µ1 and µ2 are close in the Prokhorov distance, and the curves η1 and η2 are close in the uniform distance. Le Gall / Miermont: [Le 13, Mie13] Convergence of random quadrangulations of the sphere to the Brownian map Bettinelli-Miermont: [BM15] Convergence of random quadrangulations of the disk to the Brownian disk Miller-Sheffield: [MS15a,b,c,MS16b,c,d] Construction of metric on the 8/3-LQG sphere, cone, disk which is isometric to the Brownian map, plane, disk Duplantier-Miller-Sheffield: [DMS14] General theory of quantum surfaces and conformal welding Sheffield: [She16] Basic theory of conformal welding of quantum surfaces Consequence: Convergence of self-avoiding walk on random quadrangulations to SLE 8/3 on 8/3-LQG Gwynne-Miller: Convergence of random quadrangulations of the upper half-plane to the Brownian halfplane Gwynne-Miller: [GM16a] Convergence of the discrete graph gluing of random quadrangulations of the upper halfplane to the metric gluing of Brownian halfplanes Gwynne-Miller: [GM16b] Conformal welding of 8/3-LQG surfaces is the same as the metric gluing
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