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The transience of the usual random walk on $\mathbb{Z}^3$ is a well known mathematical facts of which we know many proof, including using electrical networks, Fourier transform ... Those concepts while extremely useful are hard to communicate to someone who is not used to them and as a result make the whole explanation not very clear, plus the result is kinda obscure (Calculate something, deduce a result, but you can't really get a feel for it).

Do you know a proof that is explainable to someone with just basic notions of probability ?

Rob Arthan
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Statistic Dean
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  • Can you assume that the person knows the $p$ series test for infinite series (covered in calc II in the US)? IIRC there is a fairly straightforward argument that is based on whether $\sum_{n=1}^\infty n^{-d/2}$ is finite. – Ian Oct 14 '20 at 21:50
  • I believe this is one way to think about it that might be accessible. Would you mind sharing a complete proof as an answer ? – Statistic Dean Oct 14 '20 at 22:10

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Have a look at the Example 12.2 on page 6 of PDF at this link. This is a chapter of the freely available book Introduction to Probability by Grinstead and Snell. They describe how to use basic counting to conclude transience of random walks in dimension 3 and higher.

Kasun Fernando
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