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 $b$th 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, $LM_b$. In simply connected domains, this process is the Schramm-Loewner evolution with parameter $\kappa = 6/(2b+1)$. In non-simply connected domains, it corresponds to the harmonic random Loewner chains as introduced by Zhan.