The geodesics in Liouville quantum gravity are not Schramm–Loewner evolutions

Miller, Jason; Qian, Wei

doi:10.1007/s00440-019-00949-7

Cited by 25 publications

(33 citation statements)

References 33 publications

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“…• For each fixed z, w ∈ C, the D h -geodesic from z to w is a.s. unique. This follows from, e.g., the proof of [62,Theorem 1.2] (see also [36,Lemma 2.2]).…”

Section: Outlinementioning

confidence: 75%

“…2.5). It is shown in [62] that D h -geodesics are conformally removable and their laws are mutually singular with respect to Schramm-Loewner evolution curves. After the appearance of this paper, the work [43] proved that D h satisfies a version of the KPZ formula [27,53] and the work [1] proved a concentration result for the LQG mass of a D h -metric ball.…”

Section: Corollary 13 (Rotational Invariance)mentioning

confidence: 99%

“…By a standard Radon-Nikodym derivative calculation for the GFF (see, e.g., [62,Lemma 4.1]) and the translation and scale invariance of the law of h, modulo additive constant, for each α > 0 there is a constant C = C(α, s) > 0 such that the following is true. The conditional law given of (h −h 1+s (z)…”

Section: Local Independence For the Gffmentioning

confidence: 99%

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

2020

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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.

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“…• For each fixed z, w ∈ C, the D h -geodesic from z to w is a.s. unique. This follows from, e.g., the proof of [62,Theorem 1.2] (see also [36,Lemma 2.2]).…”

Section: Outlinementioning

confidence: 75%

Section: Corollary 13 (Rotational Invariance)mentioning

confidence: 99%

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

2020

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“…A similar definition of a strong LQG metric has appeared in earlier literature. Indeed, the paper [37] proved several properties of geodesics for any metric associated with γ -LQG which satisfies a similar list of axioms to the ones in our definition of a strong LQG metric; however, at that point such a metric had only been constructed for γ = √ 8/3. 3 It far from obvious that subsequential limits of LFPP satisfy (1.9).…”

Section: Remark 11mentioning

confidence: 77%

“…We call a metric satisfying these axioms a weak LQG metric. A closely related list of axioms appeared previously in [37]. • Properties of weak LQG metrics We prove several quantitative properties for a general weak LQG metric.…”

Section: Overviewmentioning

confidence: 99%

Weak LQG metrics and Liouville first passage percolation

Dubédat

Falconet

Gwynne

et al. 2020

Probab. Theory Relat. Fields

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For $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we define a weak$$\gamma $$ γ -Liouville quantum gravity (LQG) metric to be a function $$h\mapsto D_h$$ h ↦ D h which takes in an instance of the planar Gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding et al. (Tightness of Liouville first passage percolation for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , 2019. ArXiv e-prints, arXiv:1904.08021). It is also known that these axioms are satisfied for the $$\sqrt{8/3}$$ 8 / 3 -LQG metric constructed by Miller and Sheffield (2013–2016). For any weak $$\gamma $$ γ -LQG metric, we obtain moment bounds for diameters of sets as well as point-to-point, set-to-set, and point-to-set distances. We also show that any such metric is locally bi-Hölder continuous with respect to the Euclidean metric and compute the optimal Hölder exponents in both directions. Finally, we show that LQG geodesics cannot spend a long time near a straight line or the boundary of a metric ball. These results are used in subsequent work by Gwynne and Miller which proves that the weak $$\gamma $$ γ -LQG metric is unique for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , which in turn gives the uniqueness of the subsequential limit of Liouville first passage percolation. However, most of our results are new even in the special case when $$\gamma =\sqrt{8/3}$$ γ = 8 / 3 .

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Liouville quantum gravity and the Brownian map III: the conformal structure is determined

Miller

Sheffield⋆

2021

Probab. Theory Relat. Fields

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Previous works in this series have shown that an instance of a $$\sqrt{8/3}$$ 8 / 3 -Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the $$\sqrt{8/3}$$ 8 / 3 -LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and $$\sqrt{8/3}$$ 8 / 3 -LQG surfaces with other topologies.

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The geodesics in Liouville quantum gravity are not Schramm–Loewner evolutions

Cited by 25 publications

References 33 publications

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

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

Weak LQG metrics and Liouville first passage percolation

Liouville quantum gravity and the Brownian map III: the conformal structure is determined

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