# 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}
}
• Published 12 November 2015
• Mathematics
• arXiv: Probability
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…

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