# Bipolar orientations on planar maps and SLE$_{12}$

@article{Kenyon2015BipolarOO, title={Bipolar orientations on planar maps and SLE\$\_\{12\}\$}, author={Richard W. Kenyon and Jason Miller and Scott Sheffield and David Bruce Wilson}, journal={arXiv: Probability}, year={2015} }

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 the "peano curve" surrounding the tree of left-most paths to the sink, converges in law with respect to the peanosphere topology to a $\sqrt{4/3}$-Liouville quantum gravity surface decorated by an independent Schramm-Loewner evolution with parameter $\kappa=12$ (i.e., SLE$_{12}$). This result is…

## 51 Citations

Plane bipolar orientations and quadrant walks.

- Mathematics
- 2019

Bipolar orientations of planar maps have recently attracted some interest in combinatorics, probability theory and theoretical physics. Plane bipolar orientations with $n$ edges are known to be…

Bipolar oriented random planar maps with large faces and exotic SLE$_\kappa(\rho)$ processes

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We consider bipolar oriented random planar maps with heavy-tailed face degrees. We show for each α ∈ (1, 2) that if the face degree is in the domain of attraction of an α-stable Lévy process, the…

Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense

- Mathematics
- 2016

Kenyon, Miller, Sheffield, and Wilson (2015) showed how to encode a random bipolar-oriented planar map by means of a random walk with a certain step size distribution. Using this encoding together…

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

- MathematicsProbability Theory and Related Fields
- 2020

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…

Schnyder woods, SLE$_{(16)}$, and Liouville quantum gravity

- Mathematics
- 2017

In 1990, Schnyder used a 3-spanning-tree decomposition of a simple triangulation, now known as the Schnyder wood, to give a fundamental grid-embedding algorithm for planar maps. We show that a…

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

- 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…

On the enumeration of plane bipolar posets and transversal structures

- MathematicsTrends in Mathematics
- 2021

It is shown that plane bipolar posets and transversal structures can be set in correspondence to certain (weighted) models of quadrant walks, via suitable specializations of a bijection due to Kenyon, Miller, Sheffield and Wilson.

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…

Anomalous diffusion of random walk on random planar maps

- MathematicsProbability 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.…

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