# 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} }

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…

## 34 Citations

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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

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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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A distance exponent for Liouville quantum gravity

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- 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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Mating of trees for random planar maps and Liouville quantum gravity: a survey

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The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity

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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,…

KPZ formulas for the Liouville quantum gravity metric

- MathematicsTransactions of the American Mathematical Society
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Let $\gamma\in (0,2)$, let $h$ be the planar Gaussian free field, and let $D_h$ be the associated $\gamma$-Liouville quantum gravity (LQG) metric. We prove that for any random Borel set $X \subset…

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Random walk on random planar maps: Spectral dimension, resistance and displacement

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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

- MathematicsNotices 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

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- 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

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- 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 Tutte embedding of the mated-CRT map converges to Liouville quantum gravity

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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,…

KPZ formulas for the Liouville quantum gravity metric

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Let $\gamma\in (0,2)$, let $h$ be the planar Gaussian free field, and let $D_h$ be the associated $\gamma$-Liouville quantum gravity (LQG) metric. We prove that for any random Borel set $X \subset…

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