Bipolar orientations on planar maps and SLE12

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., 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. Keywords: Bipolar oriention; random planar map; Schramm–Loewner evolution; Liouville quantum gravity; continuum; random tree