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Every mathematician is familiar with the result (due to Pólya) that for a random walk in a $d$-dimensional lattice, the probability $p(d)$ for returning to the origin eventually is $1$ if $d=1,2$, but $<1$ if $d>2$. There are also many discussions of this result on this site (such as Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1 discussing a proof).

What I have never seen is an intuitive explanation for the extremely counterintuitive result that for dimensions $1$ and $2$ the random walk returns to the origin almost surely but for higher dimensions it does not.

Does such an intuition exist?

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user369798733
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  • In describing it as an "extremely counterintuitive result" you are somewhat ruling out 'intuitive explanations', no? Anyway, how about this: space is big but the plane isn't that big. – AakashM Mar 19 '14 at 12:50
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    The Monty-Hall Problem (aka "Three doors, two goats") is also extremely counterintuitive, yet has an intuitive explanation. As for the statement "space is big but the plane isn't that big", yes, that is clearly what the result implies, but what's missing is why that is so. In what sense is space "bigger" than the plane? – user369798733 Mar 19 '14 at 13:23
  • Fair point. I see that the proof of the expected return in two dimensions depends on the divergence of the harmonic series, which in some sense only just diverges, so maybe the plane is in fact as big a thing as you can have to still get an expected return. – AakashM Mar 19 '14 at 13:37

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I read the following intuitive statement somewhere: After N steps in any dimension, you are at average distance $sqrt(N) $ from the origin. In 3 dimensions therefore you can reach $N^{3/2}$ points, which is growing faster than N, so you cannot on average visit all the points. In 2d however this calculation gives that you can visit N points, which is (just) achievable.

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