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.