Why Do Random Walks Get Lost in 3D?

In this video, we try to gain some intuition for why symmetric random walks are recurrent in 1 and 2D, but transient in 3D. This was proved by mathematician George Pólya in 1921.

Links for further reading:

Special thanks to Ryan Adams, Elmira Amirloo, Alex Beatson, and Yair Shenfeld for feedback on this video.

Music: Trinkets by Vincent Rubinetti

Links:

If you'd like to help support the channel (completely optional), you can donate a cup of coffee via the following:

[vihnupradeep]

This channel is so terribly underrated ! Good work man 💯

[alexistorstenson]

super cool video! i'm watching this in the middle of the night after my math major friend told me about his research project involving random walks. been absolutely fascinated by the concept since i heard of it and this does a great job of breaking it down!

[tedsheridan]

Great video. This is much more elegant and easier to follow than Mathamaniac's video.

[bryanbischof]

I hadn’t previously thought of this result in concentration or measure! I think this is a good connection!

[pafloxyq]

nice one !!!...it feels so gr8 to see people using Manim to make such videos.

[rinfrinkleko]

What a great video! Thank you to share such a wonderful video that vividly explained the random walks. I was recently take research about graphs, you explaination about random walks really help me to understand it.

[MrMareczek]

Those videos are so high quality <3

[DC]

Hey, nice video. I had a doubt at 9:25, as you mentioned the expected number of returns to the origin is infinite, but how does that tell us anything about the probability of returning to the origin?

[rmschindler]

I wonder if there are different species of ‘random’.

[superevilgoldfish]

13:05 isn't clear to me. Why do you say that the expected # of steps when you first reach distance r =r^2 ? Just before you said that regardless of dimensionality, the expected distance squared is n. So I thought that you would have expected # of steps when you first reach distance r =r^0.5 instead .
I tried to follow you from a different path. If the walker is 'not drunk' and in dimension d, than to get to a distance r he needs to take r/sqrt(d) in each dimension(r^2=x^2+y^2+z^2 ->x, y, z=r/sqrt(3), total # of steps n=3*r/sqrt(3)). So the number of steps he needs to take in total to get to r is d*r/sqrt(d), which scales like sqrt(d) for a given r. You can see it diverges but I feel it's not as strong as your argument.

I hope my question is clear, I would love to understand your last argument :)

[antares]

at 10:00, is it okay to multiply the two probabilities like that? The two events aren't independent as for every step along y, you don't take one along x. I'm sure the resulting series will still diverge, but there are others factors to account for.

[the_real_amir]

Thanks for making this video. Very well made

[kwillie]

I assume this is related to why space has 3 spatial dimensions

[farouku]

Hello! I’m a freshman at pton and I was wondering which class I may be able to take so I can learn more about the proofs? ORF 309 maybe?

[АлександрСавельев-ьн]

Great video!
9:20 How does it follow from the fact that the expectation of the number of visits to the origin is infinite that the probability of returning to the origin is 1?

[Niglnws]

I understand it almost well. But i am not good at infinities mathematics and about how some infinities are greater than others. But i believe that if 3d walk given infinite steps greater than the infinity of the 2d walk, i t will return at least once with a probability reaching 1.
Before i watched the video and saw hoe you simulate the 2d walk as 2 of the 1d. I thought of something similar which is the plane of x, y and that plane go into and out of the screen or paper as the z axis with probabilities 0.5. If in the 2d walk or the xy plane we return infinite number of times, and Also the z direction as a 1d walk return to 0 infinite number of times, then they probably meet infinite number of times but here with infnity less than infinity in 2d. So i compensate the 3d by giving it a very large infinity to guarantee that the infinity of xy meet the infinity in case of 1d (z axis walk). 😶

[hellogoodbye]

What is the probability of returning in 4D?

[tianjiaowang]

Why is three simultaneous 1D walks has 8 outcomes at each steps? I thought it was 6

[rushikeshshinde]

Great Video!! I have been wondering about this question, Does walker will return if it is walking on a S^3 aka sphere with 3 dimensional surface?

[navyatayi]

For the 1-D case,
P(Sn=0) is proportional to 1/under root(n)
Now, if n tends to infinity wouldn't the probability become 0?
but when we calculate it through the indicator variable we get the probability as 1
So I am confused about this.

It would be of great help if you could clarify this doubt since I am stuck trying to understand this. Thank you for this amazing video!