Random walks have been used also to numerically solve differential equations. See e.g. [1]
[1] http://www.jstor.org/stable/3612176 "A Proof of the Random-Walk Method for Solving Laplace's Equation in 2-D"
Random Walks: the mathematics in 1 dimension
11–19 of 19 posts
Re: Random Walks: the mathematics in 1 dimension
#12One 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.
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Re: Random Walks: the mathematics in 1 dimension
#13Earlier quoted context omitted.
Wow, do you have any proofs for this? I'm especially curious about the generalized n-dimensional case.
There is some more information and references here: http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
If 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.
Re: Random Walks: the mathematics in 1 dimension
#14Overview why the question on (and need for explanation of) random walks arise: http://www.mit.edu/~kardar/teaching/projects/chemotaxis(Andr...
How much of this led to early stages of life ?
Re: Random Walks: the mathematics in 1 dimension
#15Here is a related lecture from MIT https://youtu.be/56iFMY8QW2k which mathematically proves how it is pretty much impossible to go "happy" from gambling in a club even though intuition says otherwise.
What does "happy" mean?
It's not hard to set up a bet that gives you an arbitrarily high chance of gaining money, despite an expected value of less than 1.
Re: Random Walks: the mathematics in 1 dimension
#16love this random walk..dance?.. video:
neutral dynamics
Re: Random Walks: the mathematics in 1 dimension
#17Earlier quoted context omitted.
There is some more information and references here: http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
If 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.
Here's a reference I found for one way to do it: 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.
Re: Random Walks: the mathematics in 1 dimension
#18Earlier quoted context omitted.
If 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.
Here's a reference I found for one way to do it: 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. Th…
You're a scholar and a gentleman, merci buckets
Re: Random Walks: the mathematics in 1 dimension
#19Earlier quoted context omitted.
Wow, do you have any proofs for this? I'm especially curious about the generalized n-dimensional case.
There is some more information and references here: http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
The 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