A Multi-parameter Family of Self-avoiding Walks on the Sierpiński Gasket

Abstract

In this paper, we construct a multi-parameter family of self-avoiding walks on the Sierpiński gasket. It includes the branching model, the loop-erased random walk and the loop-erased self-repelling walk. We reproduce in a unified manner the proof of the existence of the continuum limit and the self-avoiding property of the limit processes. Our limit processes include not only all the processes obtained from the previously studied self-avoiding walk models, but also the ones that have not been constructed before. While the paths of limit processes appearing in the previous works were self-avoiding or filled the whole space, our family includes continuous processes whose path is self-intersecting but does not fill the whole space.