Liouville metric of star-scale invariant fields: tails and Weyl scaling

Dubédat, Julien; Falconet, Hugo

doi:10.1007/s00440-019-00919-z

Cited by 15 publications

(25 citation statements)

References 37 publications

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“…The present article is closely related to [8,15]. In [8] it was shown that discrete Liouville first-passage percolation (shortest-path metric where the vertices are weighted by the exponential of the discrete GFF) has subsequential scaling limits for sufficiently small γ > 0.…”

Section: Two Closely-related Workmentioning

confidence: 83%

“…In [8] it was shown that discrete Liouville first-passage percolation (shortest-path metric where the vertices are weighted by the exponential of the discrete GFF) has subsequential scaling limits for sufficiently small γ > 0. On the other hand, in [15], the authors considered the case when the underlying field is a type of log-correlated Gaussian field (in the continuum) with short-range correlations (a so-called ⋆-scale invariant field) and showed that there exists a parameter γ * > 0 such that the corresponding Liouville first-passage percolation has a subsequential scaling limit for all γ < min(γ * , 0.4). The main contribution of the present article is that the result for Liouville graph distance is valid throughout the subcritical regime; i.e.…”

Section: Two Closely-related Workmentioning

confidence: 99%

“…• In both [8] and [15], the authors worked with Liouville first-passage percolation, while in the present article we work with Liouville graph distance. They are both natural approximations of LQG distances and at the moment we are equally satisfied with proving results for either of these choices (or any other reasonable choices).…”

Section: Two Closely-related Workmentioning

confidence: 99%

“…• In both [8,15], in the regime where subsequential scaling limits were proved, a crucial step of the proofs was that the box diameter has the same order as the typical distance between two points. We also note that in [10], it was shown for all γ ∈ (0, 2) that the exponents for the diameter and the typical distance are the same.…”

Section: Two Closely-related Workmentioning

confidence: 99%

“…• In [15], results regarding the Weyl scaling were also obtained. Weyl scaling is considered neither in [8] nor in the present article.…”

Section: Two Closely-related Workmentioning

confidence: 99%

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Subsequential Scaling Limits for Liouville Graph Distance

Ding

Dunlap

2020

Commun. Math. Phys.

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For 0 < γ < 2 and δ > 0, we consider the Liouville graph distance, which is the minimal number of Euclidean balls of γ-Liouville quantum gravity measure at most δ whose union contains a continuous path between two endpoints. In this paper, we show that the renormalized distance is tight and thus has subsequential scaling limits at δ → 0. In particular, we show that for all δ > 0 the diameter with respect to the Liouville graph distance has the same order as the typical distance between two endpoints.

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Section: Two Closely-related Workmentioning

confidence: 83%

Section: Two Closely-related Workmentioning

confidence: 99%

Section: Two Closely-related Workmentioning

confidence: 99%

Section: Two Closely-related Workmentioning

confidence: 99%

“…• In [15], results regarding the Weyl scaling were also obtained. Weyl scaling is considered neither in [8] nor in the present article.…”

Section: Two Closely-related Workmentioning

confidence: 99%

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Subsequential Scaling Limits for Liouville Graph Distance

Ding

Dunlap

2020

Commun. Math. Phys.

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The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

Gwynne

2020

Commun. Math. Phys.

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Let $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , let h be the planar Gaussian free field, and consider the $$\gamma $$ γ -Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$ γ -LQG metric ball with respect to the Euclidean (resp. $$\gamma $$ γ -LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$ 2 - γ d γ 2 γ + γ 2 + γ 2 2 d γ 2 (resp. $$d_\gamma -1$$ d γ - 1 ), where $$d_\gamma $$ d γ is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$ γ -LQG metric. For $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , in which case $$d_{\sqrt{8/3}}=4$$ d 8 / 3 = 4 , we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$ 8 / 3 -LQG) dimension of a $$\sqrt{8/3}$$ 8 / 3 -LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$ γ -LQG Hausdorff dimensions of the intersection of a $$\gamma $$ γ -LQG ball boundary with the set of metric $$\alpha $$ α -thick points of the field h for each $$\alpha \in \mathbb R$$ α ∈ R . Our results show that the set of $$\gamma /d_\gamma $$ γ / d γ -thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$ γ -thick points on the ball boundary has full $$\gamma $$ γ -LQG dimension.

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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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Liouville metric of star-scale invariant fields: tails and Weyl scaling

Cited by 15 publications

References 37 publications

Subsequential Scaling Limits for Liouville Graph Distance

Subsequential Scaling Limits for Liouville Graph Distance

The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

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

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