# Weak LQG metrics and Liouville first passage percolation

@article{Dubedat2020WeakLM, title={Weak LQG metrics and Liouville first passage percolation}, author={Julien Dub'edat and Hugo Falconet and Ewain Gwynne and Joshua Pfeffer and Xin Sun}, journal={Probability Theory and Related Fields}, year={2020}, pages={1-68} }

For $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we define a weak $$\gamma $$ γ - Liouville quantum gravity ( LQG ) metric to be a function $$h\mapsto D_h$$ h ↦ D h which takes in an instance of the planar Gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding et al. (Tightness of Liouville first…

## 31 Citations

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

- Mathematics
- 2019

We study Liouville first passage percolation metrics associated to a Gaussian free field h $h$ mollified by the two-dimensional heat kernel p t $p_{t}$ in the bulk, and related star-scale invariant…

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

- MathematicsInventiones mathematicae
- 2020

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…

Volume of metric balls in Liouville quantum gravity

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- 2020

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Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach

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- 2020

There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge $${{\mathbf {c}}}_{\mathrm M} \in (-\infty ,1]$$ c M ∈ ( -…

Weak Liouville quantum gravity metrics with matter central charge $\mathbf{c} \in (-\infty, 25)$

- Mathematics
- 2021

Physics considerations suggest that a theory of Liouville quantum gravity (LQG) should exist for all values of matter central charge c ∈ (−∞, 25). Probabilists have rigorously defined LQG as a random…

The geodesics in Liouville quantum gravity are not Schramm–Loewner evolutions

- MathematicsProbability Theory and Related Fields
- 2019

We prove that the geodesics associated with any metric generated from Liouville quantum gravity (LQG) which satisfies certain natural hypotheses are necessarily singular with respect to the law of…

Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense

- MathematicsElectronic Journal of Probability
- 2021

Recent works have shown that random triangulations decorated by critical ($p=1/2$) Bernoulli site percolation converge in the scaling limit to a $\sqrt{8/3}$-Liouville quantum gravity (LQG) surface…

The Tutte Embedding of the Poisson–Voronoi Tessellation of the Brownian Disk Converges to $$\sqrt{8/3}$$-Liouville Quantum Gravity

- MathematicsCommunications in Mathematical Physics
- 2019

Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a…

Geodesics and metric ball boundaries in Liouville quantum gravity

- MathematicsProbability Theory and Related Fields
- 2022

Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $\gamma \in (0,2)$. We establish a…

Geodesic networks in Liouville quantum gravity surfaces

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Recent work has shown that for $\gamma \in (0,2)$, a Liouville quantum gravity (LQG) surface can be endowed with a canonical metric. We prove several results concerning geodesics for this metric. In…

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There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge $${{\mathbf {c}}}_{\mathrm M} \in (-\infty ,1]$$ c M ∈ ( -…