  A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in
the square lattice with no self-intersection. A planar self-avoiding polygon
(SAP) is a loop with no self-intersection. In this paper we present conjectures
for the scaling limit of the uniform measures on these objects. The conjectures
are based on recent results on the stochastic Loewner evolution and
non-disconnecting Brownian motions. New heuristic derivations are given for the
critical exponents for SAWs and SAPs.
