Bipolar orientations on planar maps and SLE$_{12}$
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Authors
Kenyon, Richard
Miller, Jason
Sheffield, Scott
Wilson, David Bruce
Journal Title
Annals of Probability
Publisher
Institute of Mathematical Statistics
Type
Article
Metadata
Show full item recordCitation
Kenyon, R., Miller, J., Sheffield, S., & Wilson, D. B. Bipolar orientations on planar maps and SLE$_{12}$. Annals of Probability https://doi.org/10.17863/CAM.26729
Abstract
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 universal in the sense that it holds for bipolar-oriented triangulations, quadrangulations, $k$-angulations, and maps in which face sizes are mixed.
Sponsorship
Engineering and Physical Sciences Research Council (EP/I03372X/1)
Engineering and Physical Sciences Research Council (EP/L018896/1)
Identifiers
This record's DOI: https://doi.org/10.17863/CAM.26729
This record's URL: https://www.repository.cam.ac.uk/handle/1810/279351
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