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I just read a proof that simple symmetric random walks on discrete state spaces are recurrent in 1 and 2 dimensions but transient in 3 or more dimensions. Does anyone have some intuition as to why this is true? Examining the proof isn't helping me that much.

After reading Robert's answer (which actually resembles the proof I read!) I feel that I wasn't clear enough with my question.

In particular, is there something that is making 2 dimensions the magic number? (or 3 depending on how you look at it)

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