The Rate of Convergence of Loop-Erased Random Walk to SLE(2) (08rit136)

Objectives

One of the broad goals of statistical mechanics is to understand the behaviour of a physical system at criticality; that is, at (or near) the temperature at which a phase transition occurs. In elaborate continuous physical systems it is often useful to approximate this continuous system by a discrete, or lattice, model. These lattice models lend themselves better to simulation. Furthermore, they are often more tractable mathematically and physical or chemical predictions about the original system can be proved rigorously using results established for the lattice model. 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 contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.\'\' However, although there is knowledge of the scaling limit in some select few examples, there is essentially nothing known about the rates of convergence of any of these discrete models to SLE. In fact, this 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.\'\' (Proceedings of the ICM 2006, Vol I, page 532) Therefore, the objective of our research in teams meeting is to study the rate of convergence of loop-erased random walk to SLE(2). In our opinion, this is the most promising case and the first one that should be considered. Loop-erased random walk has been extensively studied, and there are a number of tools available for analyzing them including a detailed proof of convergence to radial SLE(2) by G. Lawler, O. Schramm, and W. Werner; a proof of convergence to chordal SLE(2) via Wilson\'s algorithm by F. Johansson; the scaling limit of Fomin\'s identity in terms of SLE(2) by M. Kozdron; and the work by C. Benes on discrete half-plane capacity, a natural discrete analogue to the half-plane capacity, which parametrizes the SLE curves. Our specific goal during a week of intensive study in Banff is to determine a nontrivial estimate (including an outline of the proof) for the speed of convergence of loop-erased random walk to SLE(2). Since the three team members work at institutions that are distant from each other, it would be crucial for us to be able to use the BIRS facility to intensify our ongoing discussion on this question and spend a full week making progress on this important question.