The Laplacian-b random walk and the Schramm-Loewner evolution

Abstract

The Laplacian-b random walk is a measure on self-avoiding paths that at each step has translation probabilities weighted by the bth power of the probability that a simple random walk avoids the path up to that point. We give a heuristic argument as to what the scaling limit should be and call this process the Laplacian-b motion, LMb. In simply connected domains, this process is the Schramm-Loewner evolution with parameter κ = 6/(2b + 1). In non-simply connected domains, it corresponds to the harmonic random Loewner chains as introduced by Zhan. ©2006 University of Illinois.