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I watched this PBS video a while ago (relevant part here) and have been trying to get my head around the idea of transient walks. The video says that a recurrent random walk is one that is guaranteed to return to it's starting position - all 1D and 2D walks - and a walk is transient if there is a positive probability that it never returns - 3D or higher. I've tried to have a think about this and looked some stuff up but I haven't had any breakthroughs.

What confuses me is this: A random walk in 3 dimensions can be split up into 3 independent random 1D walks. If each of these walks is guaranteed to return to the starting position infinitely many times we can say that there is a finite positive probability that they will return to the starting point on a given 'turn'. The product of the three finite probabilities is finite so isn't there a finite chance that any random walk in three dimensions will return to the start on any given 'turn' and hence they are guaranteed to return at some point?

I imagine I am just making incorrect assumptions about the nature of these infinite systems as is too easy to do but I'd like to know exactly where my intuition is wrong.

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  • $\begingroup$ I am not an expert, but the obvious problem seems to be your assertion that "we can say that there is a finite positive probability that they will return to the starting point on a given 'turn'" -- if "number of returns in 1D to starting point before turn $n$"/$n$ tends to zero, then your 'probability' can be 0, even if the number of returns is infinite. $\endgroup$ Commented May 1, 2017 at 9:08
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    $\begingroup$ Indeed the trouble is that the probability that the second and third coordinates are both at zero when the first coordinate is back at zero for the $n$th time, is decreasing with $n$. The argument you delineate fails if these probabilities sum to less than $1$. (Simple proofs of recurrence/transience can be reduced, considering the mean number of visits of the origin, to noticing that the series $\sum n^{-d/2}$ diverges when the dimension $d$ is $d=1$ or $2$ and converges when $d\geqslant3$... but you did not ask for a proof, did you?) $\endgroup$
    – Did
    Commented May 1, 2017 at 9:59
  • $\begingroup$ I cannot figure out what your question is. What assumptions are you making? What is your conclusion? Can you tell me step by step how you derived your conclusion and what your reasoning is? I'll give you a few facts and then you may have the answer to your own question. Maybe you'll realize what assumptions you were making that you didn't know how to explain earlier because you performed some calculations without knowing how you did it. If you later figure out the resolution to your problem and write an answer explaining it then accept your own answer which you can do 2 days after you post it, $\endgroup$
    – Timothy
    Commented Jan 29, 2020 at 4:52
  • $\begingroup$ it may be very valuable to other people who have the same question and find your answer by a Google search. Anyway, here's the fact. In 2 dimensions, think of the random walk as being continuous rather than discrete because the tendencies approach that as you approach an infinite amount of time. Also, imagine there are concentric circles around the starting point equally spaced apart. Starting on one circle, the probability that the next one you will hit other than that one itself is the the next one out is a tiny bit more than 50%. I believe I once figured out a proof I can't remember anymore $\endgroup$
    – Timothy
    Commented Jan 29, 2020 at 5:13
  • $\begingroup$ where Starting somewhere, the probability that the next time you hit a circle that's either twice or half the diameter of the one you're on, it will be the one of twice the diameter is exactly 50%. $\endgroup$
    – Timothy
    Commented Jan 29, 2020 at 5:16

2 Answers 2

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In 1 dimension, although the expected number of times that you will return to the origin before a give time approaches infinity as time approaches infinity, it varies sublinearly with time. When you have 3 independent 1-dimensional random walks all starting at the origin, although you can expect that each of them individually will return to the origin infinitely many times, it does not follow that there will ever be a time when all 3 of them are at the origin, although that might happen.

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The problem is that you don't want each of the three random walks to come back to zero on its own axis. You want those come-back to synchronize, that is, to come back to zero on every axis at the same time.

It turns out that each 1d walk comes back "sufficiently enough" to zero to allow for the synchronization in $2d$, but not in $3d$. Said differently, you're right in the fact that the axis $x_1 = 0$, $x_2 = 0$ and $x_3 = 0$ will be visited an infinite amount of time, but this is very different from visiting $(0,0,0)$ an infinite amount of time.

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