  The scaling limit of the two-dimensional self-avoiding walk (SAW) is believed
to be given by the Schramm-Loewner evolution (SLE) with the parameter kappa
equal to 8/3. The scaling limit of the SAW has a natural parameterization and
SLE has a standard parameterization using the half-plane capacity. These two
parameterizations do not correspond with one another. To make the scaling limit
of the SAW and SLE agree as parameterized curves, we must reparameterize one of
them. We present Monte Carlo results that show that if we reparameterize the
SAW using the half-plane capacity, then it agrees well with SLE with its
standard parameterization. We then consider how to reparameterize SLE to make
it agree with the SAW with its natural parameterization. We argue using Monte
Carlo results that the so-called p-variation of the SLE curve with p=1/nu=4/3
provides a parameterization that corresponds to the natural parameterization of
the SAW.
