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

@article{Gwynne2020AMA,
title={A mating-of-trees approach for graph distances in random planar maps},
author={Ewain Gwynne and Nina Holden and Xin Sun},
journal={Probability Theory and Related Fields},
year={2020},
pages={1-60}
}
• Published 2 November 2017
• Mathematics
• Probability Theory and Related Fields
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 ∈ ( 0 , 2 ) . The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; $$\gamma =\sqrt{8/3}$$ γ = 8 / 3 ); and planar maps…
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We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most $$n^{1/4 + o_n(1)}$$ n 1 / 4 + o n ( 1 ) in n units of time.
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Let $$\gamma \in (0,2)$$γ∈(0,2) and let h be the random distribution on $$\mathbb C$$C which describes a $$\gamma$$γ-Liouville quantum gravity (LQG) cone. Also let $$\kappa = 16/\gamma ^2 Weak LQG metrics and Liouville first passage percolation • Mathematics Probability Theory and Related Fields • 2020 For$$\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$$h ↦ D h which takes in an instance of the planar Mating of trees for random planar maps and Liouville quantum gravity: a survey • Mathematics • 2019 We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The Bounds for distances and geodesic dimension in Liouville first passage percolation • Mathematics Electronic Communications in Probability • 2019 For \xi \geq 0, Liouville first passage percolation (LFPP) is the random metric on \varepsilon \mathbb Z^2 obtained by weighting each vertex by \varepsilon e^{\xi h_\varepsilon(z)}, where The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds • Mathematics Communications in Mathematical Physics • 2019 We prove that for each$$\gamma \in (0,2)$$γ ∈ ( 0 , 2 ) , there is an exponent$$d_\gamma > 2$$d γ > 2 , the “fractal dimension of$$\gamma $$γ -Liouville quantum gravity (LQG)”, which describes The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity • Mathematics The Annals of Probability • 2017 We prove that the Tutte embeddings (a.k.a. harmonic/embeddings) of certain random planar maps converge to \gamma-Liouville quantum gravity (\gamma-LQG). Specifically, we treat mated-CRT maps, ## References SHOWING 1-10 OF 77 REFERENCES Random walk on random planar maps: Spectral dimension, resistance and displacement • Mathematics • 2017 We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d. increments or a two-dimensional Brownian motion via a Percolation on Triangulations, and a Bijective Path to Liouville Quantum Gravity • Mathematics Notices of the American Mathematical Society • 2019 We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of Anomalous diffusion of random walk on random planar maps • Mathematics Probability Theory and Related Fields • 2020 We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most$$n^{1/4 + o_n(1)}$$n 1 / 4 + o n ( 1 ) in n units of time. External diffusion-limited aggregation on a spanning-tree-weighted random planar map • Computer Science, Mathematics • 2019 These proofs are based on a special relationship between DLA and LERW on spanning-tree-weighted random planar maps as well as estimates for distances in such maps which come from the theory of Liouville quantum gravity. A distance exponent for Liouville quantum gravity • Mathematics • 2016 Let$$\gamma \in (0,2)$$γ∈(0,2) and let h be the random distribution on$$\mathbb C$$C which describes a$$\gamma $$γ-Liouville quantum gravity (LQG) cone. Also let$$\kappa = 16/\gamma ^2
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The Annals of Probability
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We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by
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We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by
Bounds for distances and geodesic dimension in Liouville first passage percolation
• Mathematics
Electronic Communications in Probability
• 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
The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds
• Mathematics
Communications in Mathematical Physics
• 2019
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 Tutte embedding of the mated-CRT map converges to Liouville quantum gravity
• Mathematics
The Annals of Probability
• 2017
We prove that the Tutte embeddings (a.k.a. harmonic/embeddings) of certain random planar maps converge to $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Specifically, we treat mated-CRT maps,