Monte Carlo comparisons of the self-avoiding walk and SLE as parameterized curves

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 κ 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/ν = 4/3 provides a parameterization that corresponds to the natural parameterization of the SAW.