On internet I have read multiple times that if you have a 2D random walk and the walk continous for some large amount of step, the probability is almost 1 that our walk will cross the starting point. What is the argument for that? I tried to look up some proof but they were all to complicated. How can we argue for that mathematically but simple?
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Please make clear what is the "2D random walk", best give links with (intuitive and/or precise) proofs, and related to the one or the other link point out which step is hard to understand. (I suppose we have a discrete random walk on $\Bbb Z^2$ starting in $(0,0)$, and each "step" is determined by adding one of the vectors $(1,0)$, $(-1,0)$, $(0,1)$, $(0,-1)$. And the walk "continues" in the same manner for "many steps".) – dan_fulea Nov 06 '20 at 16:55
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keywords: "Polya's random walk theorem". – user619894 Nov 06 '20 at 16:56