The Rate of Convergence of LERW to SLE(2)

In 2000, O. Schramm [4] introduced a one-parameter family of random growth processes in two dimen-sions which he called the stochastic Loewner evolution (now also called the Schramm-Loewner evolution) or SLE. In the past several years, SLE techniques have been successfully applied to analyze a variety of two-dimensional statistical mechanics models including percolation, the Ising model, the Q-state Potts model, uniform spanning trees, loop-erased random walk, and self-avoiding walk. Furthermore, SLE has provided a mathematically rigorous framework for establishing predictions made by conformal field theory (CFT), and much current research is being done to further strengthen and explain the links between SLE and CFT. The importance of proving predictions about such models made by conformal field theory in a rigorous mathematical sense was acknowledged when W. Werner was awarded a Fields medal in 2006 for “his con-tributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory. ” Although there is knowledge of the scaling limit in the aforementioned models, there is essentially nothing known about the rates of convergence of any of these discrete models to SLE. In fact, this mathematically important open problem was communicated by Schramm in his plenary lecture at the International Congress of Mathematicians in Madrid in 2006: “Obtain reasonable estimates for the speed of convergence of the discrete processes which are known to converge to SLE. ” (See [5], page 532.