Consider the
hexagonal or square lattice in the two-dimensional upper half-plane. Myopic
random walk is the following walk on the lattice from 0 to infinity: Starting
from the origin, it moves to adjacent points chosen uniformly amongst the neighbouring points that do not lead to the walk getting
trapped. It has been conjectured (by W. Werner and others) that the scaling
limit (i.e. as the mesh size of the lattice tends to 0) of myopic random walk
is SLE(6). In the hexagonal lattice, this is
already known (Smirnov, Camia and Newman) due to its
relation with percolation. We will present a proof of the convergence of
myopic random walk to SLE(6) that does not require
percolation. This is joint work with Phillip Yam. |