Tightness of Liouville first passage percolation for $\gamma \in (0,2)$

Ding, Jian; Dubédat, Julien; Dunlap, Alexander; Falconet, Hugo

doi:10.1007/s10240-020-00121-1

“…By (1.6), this results in scaling distances by C ξ/γ = C 1/d γ , which is consistent with the 2 The reason why we sometimes restrict to bounded continuous functions is to ensure that the convolution with the whole-plane heat kernel is finite (so D ε h is defined) and that the results about subsequential limits of LFPP in [16,18] are applicable. 3 One can also consider other variants of LFPP, defined using different approximations of the GFF, but we consider h * ε here since this is the approximation for which tightness is proven in [16]. If we knew tightness and some basic properties of the subsequential limiting metrics for LFPP defined using a different approximation of the GFF, then Theorem 1.8 below would show that these variants of LFPP also converge to the γ -LQG metric.…”

Section: Overviewsupporting

confidence: 56%

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

Gwynne

¹

,

Miller

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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“…The following theorem was proven in [DDDF19,GM19d,DFG+19,GM19a,GM19c,GM19b]. Theorem 1.5 (Existence and uniqueness of the LQG metric).…”

Section: Definition Of the Lqg Metricmentioning

confidence: 98%

“…There is also an earlier construction of the LQG metric in the special case when γ = √ 8/3 due to Miller and Sheffield [MS15,MS16a,MS16b]. This construction uses a completely different regularization procedure from the one in [DDDF19,GM19c] which only works for γ = √ 8/3. However, the Miller-Sheffield construction gives additional information about the √ 8/3-LQG metric which is not apparent from the construction in [DDDF19,GM19c], such as certain Markov properties for LQG metric balls and the connection to the Brownian map [Le 13,Mie13].…”

Section: Belowmentioning

confidence: 99%

“…More precisely, [GM19c, Theorem 1.2], building on the tightness result of [DDDF19] as well as the papers [GM19d,DFG+19,GM19a] shows that for each γ ∈ (0, 2), there is a unique (up to a deterministic global multiplicative constant) measurable function h → D h from D (C) to the space of continuous metrics on C which satisfies Definition 1.4 for U = C (note that this means φ in Axiom IV is required to be a complex affine map). As explained in [GM19c, Remark 1.5], this gives a way to define D h whenever h is a GFF plus a continuous function on an open domain U ⊂ C in such a way that Axioms I through II hold.…”

Section: Definition Of the Lqg Metricmentioning

confidence: 99%

“…This γ -LQG metric for general γ ∈ (0, 2) was constructed in [DDDF19,GM19c]. More precisely, [DDDF19] established the tightness of the approximating metrics and [GM19c] showed that the limiting metric is unique and scale invariant in the appropriate sense, building on results from [GM19d,DFG+19,GM19a]. See also [DD19,DF18,DD18] for earlier tightness results (preceding [DDDF19]); and [GM19b,GP19b] which establish the conformal coordinate change formula and the KPZ formula [KPZ88,DS11], respectively, for the LQG metric.…”

mentioning

confidence: 99%

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

Gwynne

¹

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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Integrability of SLE via conformal welding of random surfaces

Ang,

Holden,

Sun

2023

Comm Pure Appl Math

1

0

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We demonstrate how to obtain integrability results for the Schramm‐Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating‐of‐trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called . Our proof is built on two connections between SLE, LCFT, and mating‐of‐trees. Firstly, LCFT and mating‐of‐trees provide equivalent but complementary methods to describe natural random surfaces in LQG. Using a novel tool that we call the uniform embedding of an LQG surface, we extend earlier equivalence results by allowing fewer marked points and more generic singularities. Secondly, the conformal welding of these random surfaces produces SLE curves as their interfaces. In particular, we rely on the conformal welding results proved in our companion paper Ang, Holden and Sun (2023). Our paper is an essential part of a program proving integrability results for SLE, LCFT, and mating‐of‐trees based on these two connections.

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Tightness of Liouville first passage percolation for $\gamma \in (0,2)$

Cited by 58 publications

References 68 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 $$\gamma \in (0,2)$$

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

Integrability of SLE via conformal welding of random surfaces

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