Self-avoiding processes and self-repelling processes on fractals

我们分别在预sierpinski垫片和一维欧几里德空间上构造了一个自排斥行走族，它在简单随机行走和自回避行走之间连续插值，是一个参数为u的单参数族，u=0对应自回避行走；对于简单随机游动，u=1，对于自排斥游动，u<1。用位移指数和迭代对数定律得到了游动的渐近行为。结果可以进一步扩展到自吸引步行，与u bbbb1。我们的方法是基于重整化群的，我们发现使用这种方法可以构造更一般的随机链。用并行的方法得到了其渐近行为。研究了用重整化群法构造的随机链的递推性，得到了递推性的一个充分条件。特别是，我们证明了上述自我排斥和自我吸引的散步家族是反复出现的，如果你b> 0。我们还证明了存在一个正常数c>1，使得对原点的期望返回时间在0<u<c时是无穷大的。这意味着在Sierpinski垫片和一维欧几里德空间的无限长路径空间上存在唯一的，有限的，遍历不变测度。

We constructed a family of self-repelling walks on the pre-Sierpinski gasket and on the 1-dimensional Euclidean space, respectively, which continuously interpolates between the simple random walk and a self-avoiding walk It is a one-parameter family with parameter u, and u=0 corresponds to a self avoiding walk, u=1 to the simple random walk and 0<u<1 to self-repelling walks The asymptotic behaviors of the walks have been obtained in terms of displacement exponents and a law of iterated logarithms. The result can further be extended to self-attracting walks, with u>1. Our method is based on renormalization group and we found that we can construct more general stochastic chains, using this method. The asympotitic behaviors are obtained in a parallel manner.We studied also the recurrence of the stochastic chains constructed by renormalization group method and obtained a sufficient condition for recurrence. In particular, we proved the above mentioned family of self-repelling and self-attracting walks are recurrent if u>0. We also proved that there is a positive constant c>1 such that the expected return time to the origin is infinite for 0<u<c. This implies that there is a unique, sigma-finite, ergodic invariant measure on the infinite-length path space on the Sierpinski gasket and the 1-dimensional Euclidean space.