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

1.对于一类具有几乎必然Hausdorff维数D（通常随机结构的Hausdorff维数是几乎必然确定的）且D维Hausdorff测度为零的多型随机结构，得到了关于精确Hausdorff维数的一般定理.这个定理确定了给出正的和有限的Hausdorff测度的维数函数。我们将该定理应用于d维Sierpinski垫片上的自避免过程的样本路径的轨迹，这些样本路径可以被认为是随机分形.通过对预Sierpinski垫片上自排斥行走的标度极限的研究，构造了Sierpinski垫片上自排斥过程的单参数族.我们证明了我们的模型连续插值的布朗运动和Sierpinski垫片上的自避免过程。也就是说，我们证明了该过程在律收敛意义下在参数上是连续的，并且决定该过程速度的指数在参数上也是连续的。我们还建立了我们的自排斥过程的重对数律。我们的方法也可以应用到一维欧氏空间，我们得到了一类新的一维自排斥过程。

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.