The loop-erased random walk (LERW) in two dimensions is obtained by erasing loops from simple random walk. It was shown by Schramm, Werner, and myself that the scaling limit of this path is a Schramm-Loewner evolution (SLE). This result requires a special parametrization of the curve in terms of "capacilty". However, for many applications one would prefer to scale the curve in a more natural way by the (normalized) number of steps of the path. I will discuss work in the last few years that has made this "natural parametrization" rigorous. This includes joint work with Fredrik Viklund, Christian Benes, and Mohammad Rezaei.
Biography
Greg Lawler is the George Wells Beadle Distinguished Service Professor in Mathematics and Statistics at the University of Chicago after previously holding positions at Duke and Cornell Universities. He received his BA in 1976 from the University of Virginia and his Ph.D. in mathematics from Princeton University in 1979. His research is in fine properties of random walk and Brownian motion with an emphasis on problems arising in statistical physics. He is a member of the US National Academy of Sciences and of the American Academy of Arts and Sciences and was a co-recipient of the 2006 George Polya prize of the Society for Industrial and Applied Mathematics.