Bipolar orientations on planar maps and SLE₁₂

2015/11/12 by Kenyon, Richard, Miller, Jason, Sheffield, Scott +1 · 5 citations
#60J67 #82B20 #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1511.04068

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 √(4/3)-Liouville quantum gravity surface decorated by an independent Schramm-Loewner evolution with parameter κ=12 (i.e., SLE₁₂). 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.

Cited by

• Liouville quantum gravity: from random planar maps to conformal field theory
• The p-θ relation in mating of trees
• The 3-state Potts model on planar triangulations: explicit algebraic solution
• Asymptotic Face Distributions in Random Reduced \mathfrak s\mathfrak l₃ Webs
• Existence and uniqueness of the conformally covariant geodesic metric on non-simple conformal loop ensemble gaskets

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