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Bipolar orientations on planar maps and \(\mathrm{SLE}_{12}\). (English) Zbl 1466.60170
Summary: 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.

MSC:
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
05C30 Enumeration in graph theory
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