  It is widely believed that the scaling limit of self-avoiding walks (SAWs) at
the critical temperature is (i) conformally invariant, and (ii) describable by
Schramm-Loewner Evolution (SLE) with parameter $\kappa = 8/3.$ We consider SAWs
in a rectangle, which originate at its centre and end when they reach the
boundary. We assume that the scaling limit of SAWs is describable by ${\rm
SLE}_\kappa,$ with the value of $\kappa$ to be determined. It has previously
been shown by Guttmann and Kennedy \cite{GK13} that, in the scaling limit, the
ratio of the probability that a SAW hits the side of the rectangle to the
probability that it hits the end of the rectangle, depends on $\kappa.$ By
considering rectangles of fixed aspect ratio 2, and also rectangles of aspect
ratio 10, we calculate the probabilities exactly for larger and larger
rectangles. By extrapolating this data to infinite rectangle size, we obtain
the estimate $\kappa = 2.66664 \pm 0.00007$ for rectangles of aspect ratio 2
and $\kappa = 2.66675 \pm 0.00015$ for rectangles of aspect ratio 10. We also
provide numerical evidence supporting the conjectured distribution of SAWs
striking the boundary at various points in the case of rectangles with aspect
ratio 2.
