Abstract
We show that for each gamma in (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 {mathbb {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} (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 in (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}, our metric coincides with the sqrt{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), 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.
Highlights
Knowing that the Dh,A-length of every Dh geodesic is a constant times its Dh-length does not immediately imply that Dh is equal to a constant times Dh,A. This is because if Pn is a sequence of paths which converge uniformly to P, it is not necessarily true that len(Pn; Dh,A) converges to len(P; Dh,A). For this and other reasons, we will argue in a somewhat different manner than we have indicated above, though our arguments will still be based on the bi-Lipschitz equivalence of metrics and approximate independence statements for the local behavior of a geodesic at different times
We will prove Theorem 1.1 and 1.2 simultaneously by establishing a uniqueness statement for metrics under a weaker list of axioms, which are satisfied for both the strong Liouville quantum gravity (LQG) metrics considered in Sect. 1.2 and for subsequential limits of Liouville first passage percolation (LFPP)
Remark 1.12 (Proof for strong LQG metrics) As explained above, we prove Theorem 1.9 instead of just proving Theorem 1.2 since subsequential limits of LFPP are only known to be weak LQG metrics, not strong LQG metrics
Summary
To state our list of axioms precisely, we will need some preliminary definitions concerning metric spaces. Note that this is well-defined even though hU does not extend continuously to ∂U , since the definition of Dh|U involves only paths contained in U It is seen from Axioms II (locality) and III (Weyl scaling) that DhU is a measurable function of hU : if we are given an open set V ⊂ U with. By Axiom III, we can define the metric Dh in the case when h = hU + f is eξ a zero-boundary GFF plus a f DhU It is shown in [37] that continuous this metric function on U , namely Dh := satisfies a conformal coordinate change relation analogous to the one satisfied by the γ -LQG measure (as discussed just below (1.4)).
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