2021
DOI: 10.1007/s00440-021-01026-8
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Liouville quantum gravity and the Brownian map III: the conformal structure is determined

Abstract: 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 impli… Show more

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Cited by 71 publications
(97 citation statements)
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References 26 publications
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“…Note that Assumption 1.1 was shown to hold in the case γ = √ 8/3 in [29][30][31]. After the present article, it was established in [16,18] that there exists a unique metric satisfying (an equivalent form of) Assumption 1.1 for each γ ∈ (0, 2).…”
Section: Introductionmentioning
confidence: 90%
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“…Note that Assumption 1.1 was shown to hold in the case γ = √ 8/3 in [29][30][31]. After the present article, it was established in [16,18] that there exists a unique metric satisfying (an equivalent form of) Assumption 1.1 for each γ ∈ (0, 2).…”
Section: Introductionmentioning
confidence: 90%
“…Prior to this work, the metric space structure for LQG had only been constructed for γ = √ 8/3 in [28][29][30][31]. In this case, the resulting metric measure space is equivalent to that of a Brownian surface, the Gromov-Hausdorff scaling limit of uniformly random planar maps.…”
Section: Introductionmentioning
confidence: 99%
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“…The corresponding area and boundary length measures exist for all γ ∈ (0, 2] and can be rigorously defined via a regularization procedure [15,23,25]. The metric has also been defined for γ = √ 8/3 in [34][35][36] and for all γ ∈ (0, 2) in [11,21]. We also recall that an LQG surface (with or without marked points) can be viewed (and this is the perspective we will use in the present paper) as an equivalence class of domains (equipped with such an area measure, or equivalently with an instance of a variant of the Gaussian free field) under conformal maps.…”
Section: Quantum Surfacesmentioning
confidence: 99%
“…This paper is the first attempt where optimal strategy of a fish is determined through a new Feynman-type path integral approach where relative velocity of fish is on √ 8∕3-LQG surface. When this surface takes the value of √ 8∕3 , it glues to a Brownian surface (Sheffield 2007;Duplantier and Sheffield 2011;Miller and Sheffield 2016b, a;Sheffield et al 2016). The advantage of this method is that instead of using the properties of Brownian surface, one can replace it by a smooth function to do metric gluing.…”
Section: Discussionmentioning
confidence: 99%