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… 
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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
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
TLDR
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
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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KPZ formulas for the Liouville quantum gravity metric
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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