Conformal welding of uniform random trees

Abstract

A conformally balanced tree is an embedding of a given planar map into the plane with constraints on the harmonic measure of its edges such that the resulting set is unique up to scale and rotation. Bishop (2013) showed that there exists a conformal map from the exterior of the disc to the complement of such a tree. The preimage of the tree under the map is a conformal welding map which induces a lamination of the unit circle that corresponds exactly to the encoding of the tree as an excursion. We consider the distributional limits of the maps for the uniform measure on random walk excursions as the number of steps goes to infinity, normalized by conformal radius, and we show that subsequential limits are almost surely nontrivial. Additionally, we investigate the properties of laminations resulting from consistent distributions on ordered trees with variable edge length. Burdzy, Pal, and Swanson (2010) considered solid spheres of small radius reflecting in the unit interval with mass being added to the system from the left at constant rate, killed when reaching the right boundary. By transforming to a system with zero-width particles moving as independent Brownian motion, they derived a limiting stationary distribution for a particular initial distribution, as the width of a particle decreases to zero and the number of particles increases to infinity. This space-removing transformation has a direct analogy in the isomorphism between a new unbounded-range exclusion process and a superimposition of random walks with random boundary. We derive the hydrodynamic limit for these isomorphic processes.