Self-avoiding process on the Sierpinski gasket

We construct a self-avoiding process taking values in the finite Sierpinski gasket, and study its properties. We then study "continuum limit" processes that are suggested by the statistical mechanics of self-avoiding paths on the pre-Sierpinski gasket. We prove that there are three types of continuum limit processes according to the parameters defining the statistical mechanics of self-avoiding paths: (i) the self-avoiding process we construct in this paper; (ii) a deterministic motion along a "Peano curve" on the finite Sierpinski gasket; (iii) a deterministic motion along a line segment.