5 ms·
One might ask the question: what is the probability that you will return to your starting position over the course of an infinite random walk? On a 1 dimension
by Xcelerate 11y ago
One might ask the question: what is the probability that you will return to your starting position over the course of an infinite random walk? On a 1 dimensional or 2 dimensional lattice, that probability is 1. What's crazy though is that for a 3D lattice, the probability is not 1 — it's about 0.3405.
- iaw 11y agoWow, do you have any proofs for this? I'm especially curious about the generalized n-dimensional case.
- hemmer 11y agoThere is some more information and references here: http://mathworld.wolfram.com/PolyasRandomWalkConstants.html http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
- sdoering 11y agoThanks a lot. Had to save this for further reference the moment I saw it.
- Ntrails 11y agoIf you find it GP, I'd love to see where the integral constructed comes from, since that's the clever part rather than the evaluation.
- emfree 11y agoHere's a reference I found for one way to do it: http://www.math.nus.edu.sg/~matsr/ProbII/Lec6.pdf http://www.math.nus.edu.sg/~matsr/ProbII/Lec6.pdf (Theorem 2.1). You define the Green's function G(x, y) = \sum_n Pr_x(S_n=y), where x and y are 3-vectors and Pr_x(S_n=y) is the probability that an n-step random walk starting at x ends up at y. If you have an infinite random walk starting at 0, then G(0, 0) is the expected number of times that the walk returns to 0. That's what the mathworld link calls u(3). You can use Fourier inversion to compute G(0, 0) -- the link gives the gnarly details. It's pretty cool.
- Ntrails 11y agoYou're a scholar and a gentleman, merci buckets
- ccvannorman 11y agoThe probability of returning to the origin follows a very smooth logarithmic curve for dimensions 3 - 8 [copied values from the mathworld link] image: http://imgur.com/CL8MXej http://imgur.com/CL8MXej
- tristanj 11y agoI really love this proof. It's a great example of using maths to prove a counter-intuitive result. They way to prove it is rather clever, and made me appreciate what mathematicians do a lot more. Shame I've never seen it shared online. I was actually hoping the submitted article was a proof of this, but you can't have everything in life.
- deleted 11y ago[deleted]