Some intersection properties of random walk paths

@article{Erdos1960SomeIP,
  title={Some intersection properties of random walk paths},
  author={Paul L. Erdos and Stephen Taylor},
  journal={Acta Mathematica Academiae Scientiarum Hungarica},
  year={1960},
  volume={11},
  pages={231-248}
}
We complete the solution of this problem in Section 3 . Clearly, there is no problem for d-== I or 2 . The solution takes a different form in the cases d= 3, d-4, and d = 5. For example, if d = 4, an interesting consequence of the result is that, with probability 1, there are infinitely many n for which 17+ (0, n) and /1 4(2n, x ) have a point in common . This in turn implies that any two independent random walks in 4-space have infinitely many points in common . This at first surprised us… Expand
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Consider the lattice formed by all points whose coordinates are integers in d-dimensional Euclidean space, and let a point S j(n) perform a move randomly on this lattice according to the followingExpand
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  • Acta Sci. Math . Szeged
  • 1950