peanosphere

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