Oriented planar maps and annular mating of trees

Abstract

We study a class of oriented annular planar maps which are closely related to bipolar-oriented maps. We prove that such maps can be encoded by a 2D lattice walk which converges to a 2D Brownian path in the scaling limit. We further prove a mating-of-trees result for the annulus, where the aforementioned Brownian path and a generalization encodes an LQG annulus decorated by a counterclockwise space-filling SLE loop with parameter $\gamma =16{\kappa }^{-1/2}\in (0,2)$.

56 pages, 24 figures