We study self-avoiding paths on the three-dimensional pre-Sierpinski gasket. We prove the existence of the limit distribution of the scaled path length, the exponent for the mean square displacement, and the continuum limit. We also prove that the continuum-limit process is a self-avoiding process on the three-dimensional Sierpinski gasket, and that a path almost surely has infinitely fine creases.
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Kumiko Hattori, Tetsuya Hattori, Shigeo Kusuoka, Self-avoiding Paths on the Three Dimensional Sierpinski Gasket. Publ. Res. Inst. Math. Sci. 29 (1993), no. 3, pp. 455–509DOI 10.2977/PRIMS/1195167053