The Annals of Probability

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

Richard Kenyon, Jason Miller, Scott Sheffield, and David B. Wilson

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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.

Article information

Source
Ann. Probab., Volume 47, Number 3 (2019), 1240-1269.

Dates
Received: November 2016
Revised: January 2018
First available in Project Euclid: 2 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1556784019

Digital Object Identifier
doi:10.1214/18-AOP1282

Mathematical Reviews number (MathSciNet)
MR3945746

Zentralblatt MATH identifier
07067269

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 05C30: Enumeration in graph theory

Keywords
Bipolar oriention random planar map Schramm–Loewner evolution Liouville quantum gravity continuum random tree

Citation

Kenyon, Richard; Miller, Jason; Sheffield, Scott; Wilson, David B. Bipolar orientations on planar maps and $\mathrm{SLE}_{12}$. Ann. Probab. 47 (2019), no. 3, 1240--1269. doi:10.1214/18-AOP1282. https://projecteuclid.org/euclid.aop/1556784019


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