  We outline a strategy for showing convergence of loop-erased random walk on
the Z^2 square lattice to SLE(2), in the supremum norm topology that takes the
time parametrization of the curves into account. The discrete curves are
parametrized so that the walker moves at a constant speed determined by the
lattice spacing, and the SLE(2) curve has the recently introduced natural time
parametrization. Our strategy can be seen as an extension of the one used by
Lawler, Schramm, and Werner to prove convergence modulo time parametrization.
The crucial extra step is showing that the expected occupation measure of the
discrete curve, properly renormalized by the chosen time parametrization,
converges to the occupation density of the SLE(2) curve, the so-called SLE
Green's function. Although we do not prove this convergence, we rigorously
establish some partial results in this direction including a new loop-erased
random walk estimate.
