The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball
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Communications in
Mathematical Physics
The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball Ewain Gwynne University of Cambridge, Cambridge, UK. E-mail: [email protected] Received: 16 December 2019 / Accepted: 6 March 2020 © The Author(s) 2020
Abstract: Let γ ∈ (0, 2), let h be the planar Gaussian free field, and consider the γ Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundaryof a γ -LQG metric ball with γ γ γ2 2 respect to the Euclidean (resp. γ -LQG) metric is 2 − dγ γ + 2 + 2d 2 (resp. dγ − 1), γ
where dγ is the Hausdorff dimension of the whole plane with respect to the γ -LQG √ metric. For γ = 8/3, in which case d√8/3 = 4, we get that the essential supremum √ √ of Euclidean (resp. 8/3-LQG) dimension of a 8/3-LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and γ -LQG Hausdorff dimensions of the intersection of a γ -LQG ball boundary with the set of metric α-thick points of the field h for each α ∈ R. Our results show that the set of γ /dγ -thick points on the ball boundary has full Euclidean dimension and the set of γ -thick points on the ball boundary has full γ -LQG dimension. Contents 1.
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 1.2 Hausdorff dimension of an LQG metric ball boundary 1.3 Thick points on the boundary of an LQG metric ball . 1.4 Definition of the LQG metric . . . . . . . . . . . . . 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Basic notation . . . . . . . . . . . . . . . . . . . . . Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . 2.1 The one-point upper bound . . . . . . . . . . . . . . 2.2 Proofs of Hausdorff dimension upper bounds . . . . 2.3 Generalized upper bound . . . . . . . . . . . . . . . Outline of the Lower Bound Proof . . . . . . . . . . . . .
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E. Gwynne
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Short-Range Events . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definition of short-range events . . . . . . . . . . . . . . . 4.2 One-point and two-point estimates for short-range events . 4.3 Deterministic estimates truncated on the short-range event 5. Long-Range Events . . . . . . . . . . . . . . . . . . . . . . . 5.1 One-point estimate . . . . . . . . . . . . . . . . . . . . . 5.2 Two-point estimate . . . . . . . . . . . . . . . . . . . . . 5.3 Lower bounds for Hausdorff dimension . . . . . . . . . . 6. One-Point Estimate for the Event at a Single Scale . . . . . . . 6.1 Comparison of distances in annuli with positive probability 6.2 Bounds for internal diameters of annuli . . . . . . . . . . . 6.3 Uniform bounds for Radon–Nikodym derivatives . . . . . 6.4 Proof of Proposition 4.8 . . . . . . . . . . . . . . . . . . . A. Gaussia
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