# Bounds for distances and geodesic dimension in Liouville first passage percolation

@article{Gwynne2019BoundsFD, title={Bounds for distances and geodesic dimension in Liouville first passage percolation}, author={Ewain Gwynne and Joshua Pfeffer}, journal={Electronic Communications in Probability}, year={2019} }

For $\xi \geq 0$, Liouville first passage percolation (LFPP) is the random metric on $\varepsilon \mathbb Z^2$ obtained by weighting each vertex by $\varepsilon e^{\xi h_\varepsilon(z)}$, where $h_\varepsilon(z)$ is the average of the whole-plane Gaussian free field $h$ over the circle $\partial B_\varepsilon(z)$. Ding and Gwynne (2018) showed that for $\gamma \in (0,2)$, LFPP with parameter $\xi = \gamma/d_\gamma$ is related to $\gamma$-Liouville quantum gravity (LQG), where $d_\gamma$ is the…

## 22 Citations

The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds

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We prove that for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , there is an exponent $$d_\gamma > 2$$ d γ > 2 , the “fractal dimension of $$\gamma $$ γ -Liouville quantum gravity (LQG)”, which describes…

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

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Let $\gamma \in (0,2)$, let $h$ be the planar Gaussian free field, and consider the $\gamma$-Liouville quantum gravity (LQG) metric associated with $h$. We show that the essential supremum of the…

The distance exponent for Liouville first passage percolation is positive

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Discrete Liouville first passage percolation (LFPP) with parameter $\xi > 0$ is the random metric on a sub-graph of $\mathbb Z^2$ obtained by assigning each vertex $z$ a weight of $e^{\xi h(z)}$,…

Weak LQG metrics and Liouville first passage percolation

- MathematicsProbability Theory and Related Fields
- 2020

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…

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…

Comparison of discrete and continuum Liouville first passage percolation

- MathematicsElectronic Communications in Probability
- 2019

Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the conjectural $\gamma$-Liouville quantum gravity (LQG) metric, obtained by exponentiating the…

A mating-of-trees approach for graph distances in random planar maps

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We introduce a general technique for proving estimates for certain random planar maps which belong to the $$\gamma $$ γ -Liouville quantum gravity (LQG) universality class for $$\gamma \in (0,2)$$ γ…

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

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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 ∈ ( -…

External diffusion-limited aggregation on a spanning-tree-weighted random planar map

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These proofs are based on a special relationship between DLA and LERW on spanning-tree-weighted random planar maps as well as estimates for distances in such maps which come from the theory of Liouville quantum gravity.

Liouville dynamical percolation

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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…

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Let $$\{\eta (v): v\in V_N\}$${η(v):v∈VN} be a discrete Gaussian free field in a two-dimensional box $$V_N$$VN of side length N with Dirichlet boundary conditions. We study the Liouville first…

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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…

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Let $\{\eta(v): v\in V_N\}$ be a discrete Gaussian free field in a two-dimensional box $V_N$ of side length $N$ with Dirichlet boundary conditions. We study the Liouville first passage percolation,…

Comparison of discrete and continuum Liouville first passage percolation

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- 2019

Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the conjectural $\gamma$-Liouville quantum gravity (LQG) metric, obtained by exponentiating the…