Weak LQG metrics and Liouville first passage percolation

Dubédat, Julien; Falconet, Hugo; Gwynne, Ewain; Pfeffer, Joshua; Sun, Xin

doi:10.48550/arxiv.1905.00380

Cited by 4 publications

(10 citation statements)

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“…where h * ε is a particular mollified version of h obtained by integrating against the heat kernel, dγ is the dimension of γ-LQG [10,25], and the infimum is taken over all piecewise continuously differentiable paths from z to w. Ding, Dubédat, Dunlap and Falconet [7] proved that for all γ ∈ (0, 2) the laws of the suitably rescaled metrics D ε h are tight, so subsequential limits exist as ε → 0 (see also the earlier tightness results [9,12,8]). Building on this and several other works [13,20,22], Gwynne and Miller [21] showed that all subsequential limits agree and satisfy a natural list of axioms uniquely characterizing the LQG metric. So it makes sense to speak of the LQG metric D h .…”

Section: Introductionmentioning

confidence: 62%

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Volume of metric balls in Liouville quantum gravity

Ang¹,

Falconet²,

Sun³

2020

Preprint

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We study the volume of metric balls in Liouville quantum gravity (LQG). For γ ∈ (0, 2), it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly for p ∈ (−∞, 4/γ 2 ). Here, we prove that the LQG volume of LQG metric balls admits all finite moments. This answers a question of Gwynne and Miller and generalizes a result obtained by Le Gall for the Brownian map, namely, the γ = 8/3 case. We use this moment bound to show that on a compact set the volume of metric balls of size r is given by r dγ +or (1) , where dγ is the dimension of the LQG metric space. Using similar techniques, we prove analogous results for the first exit time of Liouville Brownian motion from a metric ball. Our result implies that the metric measure space structure of γ-LQG determines its conformal structure.

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Section: Introductionmentioning

confidence: 62%

“…Recently, a metric for LQG was constructed and characterized in [7,21], relying on [10,13,22,20,36]. It is also the limit of an approximation scheme similar to the one of the LQG measure.…”

Section: Lqg Metricmentioning

confidence: 99%

Volume of metric balls in Liouville quantum gravity

Ang¹,

Falconet²,

Sun³

2020

Preprint

Self Cite

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“…We also remark that the metric (i.e., distance function) for LQG was first constructed in the case that γ = 8/3 in [35,39,40] and recently for all γ ∈ (0, 2) in [7,20,9,17,19,18].…”

Section: 1mentioning

confidence: 99%

Uniqueness of the Welding Problem for Sle and Liouville Quantum Gravity

McEnteggart¹,

Miller²,

Qian³

2019

J. Inst. Math. Jussieu

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We give a simple set of geometric conditions on curves η, η in H from 0 to ∞ so that if ϕ : H → H is a homeomorphism which is conformal off η with ϕ(η) = η then ϕ is a conformal automorphism of H. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if η is a non-space-filling SLEκ curve in H from 0 to ∞, and ϕ is a homeomorphism which is conformal on H \ η, and ϕ(η), η are equal in distribution, then ϕ is a conformal automorphism of H. Applying this result for κ = 4 establishes that the welding operation for critical (γ = 2) Liouville quantum gravity (LQG) is well-defined. Applying it for κ ∈ (4, 8) gives a new proof that the welding of two independent κ/4-stable looptrees of quantum disks to produce an SLEκ on top of an independent 4/ √ κ-LQG surface is well-defined.

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“…Following [56,24,55,25,26], the Liouville first passage percolation distance D ξ, px, yq between two points x and y is given for ξ ą 0 in terms of the regularized Gaussian free field φ by D ξ, px, yq " inf…”

Section: Liouville First Passage Percolationmentioning

confidence: 99%

Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity

Barkley¹,

Budd²

2019

Class. Quantum Grav.

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Two-dimensional quantum gravity, defined either via scaling limits of random discrete surfaces or via Liouville quantum gravity, is known to possess a geometry that is genuinely fractal with a Hausdorff dimension equal to 4. Coupling gravity to a statistical system at criticality changes the fractal properties of the geometry in a way that depends on the central charge of the critical system. Establishing the dependence of the Hausdorff dimension on this central charge c has been an important open problem in physics and mathematics in the past decades. All simulation data produced thus far has supported a formula put forward by Watabiki in the nineties. However, recent rigorous bounds on the Hausdorff dimension in Liouville quantum gravity show that Watabiki's formula cannot be correct when c approacheś 8. Based on simulations of discrete surfaces encoded by random planar maps and a numerical implementation of Liouville quantum gravity, we obtain new finite-size scaling estimates of the Hausdorff dimension that are in clear contradiction with Watabiki's formula for all simulated values of c P p´8, 0q. Instead, the most reliable data in the range c P r´12.5, 0q is in very good agreement with an alternative formula that was recently suggested by Ding and Gwynne. The estimates for c P p´8,´12.5q display a negative deviation from the latter formula, but the scaling is seen to be less accurate in this regime.

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Weak LQG metrics and Liouville first passage percolation

Cited by 4 publications

References 0 publications

Volume of metric balls in Liouville quantum gravity

Volume of metric balls in Liouville quantum gravity

Uniqueness of the Welding Problem for Sle and Liouville Quantum Gravity

Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity

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