Self-avoiding walk on the high-dimensional Sierpinski gaskets and random trees

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1. We obtained a general theorem on exact Hausdorff dimensions for a class of multi-type random constructions with almost sure Hausdorff dimension D ( usually, Hausdorff dimensions of random constructions are determined almost surely ) and with zero D-dimensional Hausdorff measure. This theorem determines dimension functions which give positive and finite Hausdorff measures. We applied our theorem to the trajectories of sample paths of self-avoiding processes on the d-dimensional Sierpinski gasket, which can be considered as random fractals.2. We constructed a one-parameter family of self-repelling processes on the Sierpinski gasket, by taking scaling limits of self-repelling walks on the pre-Sierpinski gaskets. We proved that our model interpolates continuously the Brownian motion and a self-avoiding process on the Sierpinski gasket. Namely, we proved that the process is continuous in the parameter in the sense of convergence in law, and that the exponent that determines the speed of the process is also continuous in the parameter. We also established a law of the iterated logarithm of our self-repelling processes. Our approach can be applied also to one-dimensional Euclidean space and we obtained a new class of one-dimensional self-repelling processes.