Some problems concerning the structure of random walk paths

@article{Erdos1963SomePC,
  title={Some problems concerning the structure of random walk paths},
  author={Paul L. Erdos and Stephen Taylor},
  journal={Acta Mathematica Academiae Scientiarum Hungarica},
  year={1963},
  volume={11},
  pages={137-162}
}
1. In t roduct ion . We restrict our consideration to symmetric random walk, defined in the following way. Consider the lattice formed by the points of d-dimensional Euclidean space whose coordinates are integers, and let a point S,~(n) perform a move randomly on this lattice according to the rules: at time zero it is at the origin and if at any time n-1 ( n ~ l, 2, . ..) it is at some point S of the lattice, then at time n it will be at one of the 2 d lattice points nearest S, the probability… 
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