UNIFORM SPANNING TREES ON SIERPI NSKI GRAPHS

Abstract

We study spanning trees on Sierpi nski graphs (i.e., nite approxima- tions to the Sierpi nski gasket) that are chosen uniformly at random. We prove existence of a limit measure and derive a number of structural results, for instance on the degree distribution. The connection between uniform spanning trees and loop-erased random walk is then exploited to prove convergence of the latter to a continuous stochastic process akin to the Brownian motion or the Schramm- Loewner evolution. Some geometric properties of this limit process, such as the Hausdor dimension, are investigated as well. The method is also applicable to other self-similar graphs with a sucient degree of symmetry.