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

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.