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May 2019 Bipolar orientations on planar maps and $\mathrm{SLE}_{12}$
Richard Kenyon, Jason Miller, Scott Sheffield, David B. Wilson
Ann. Probab. 47(3): 1240-1269 (May 2019). DOI: 10.1214/18-AOP1282

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., $\mathrm{SLE}_{12}$). 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.

Citation

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Richard Kenyon. Jason Miller. Scott Sheffield. David B. Wilson. "Bipolar orientations on planar maps and $\mathrm{SLE}_{12}$." Ann. Probab. 47 (3) 1240 - 1269, May 2019. https://doi.org/10.1214/18-AOP1282

Information

Received: 1 November 2016; Revised: 1 January 2018; Published: May 2019
First available in Project Euclid: 2 May 2019

zbMATH: 07067269
MathSciNet: MR3945746
Digital Object Identifier: 10.1214/18-AOP1282

Subjects:
Primary: 05C30, 28C20, 60J67, 82B20

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 3 • May 2019
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