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

TL;DR

This paper establishes a universal scaling limit for bipolar-oriented planar maps, showing they converge to a Liouville quantum gravity surface decorated by SLE$_{12}$, revealing deep connections between combinatorics, probability, and geometry.

Contribution

It introduces bijections between bipolar-oriented maps and random walks, and proves their convergence to SLE$_{12}$ decorated Liouville quantum gravity surfaces, extending to various map types.

Findings

Convergence of bipolar-oriented maps to SLE$_{12}$-decorated quantum surfaces.

Universality of the scaling limit across different map types.

Bijections linking planar maps to random walks.

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., 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.