6
$\begingroup$

If you are in an infinite maze of squares, where each border between two squares has a $p$% chance of being a wall, what is the probability $F(p)$ that you are trapped in a finite space?

For a better math definition of the problem:

First imagine a finite $n \times n$ grid of squares, where $n$ is odd. The border between two squares or between a square and the outside has a $p$% chance of being a wall. You start in the middle. $f(n,p)$ is the probability that you are trapped in the $n \times n$ maze. $F(p)=\lim_{n\to\infty}f(n,p)$. What is $F(p)$, or at least what can we know about it?

$\endgroup$
5
  • $\begingroup$ Welcome to Puzzling, take our tour! Could you please provide proper attribution for this question? Or did you come up with it yourself? $\endgroup$
    – bobble
    Jan 18, 2023 at 4:53
  • 2
    $\begingroup$ Exactly this question is the subject of the field of percolation: en.wikipedia.org/wiki/Percolation_theory. It turns out that F(p) is 0 for p<1/2 and 1 for p>=1/2. $\endgroup$
    – Gareth McCaughan
    Jan 18, 2023 at 12:24
  • 3
    $\begingroup$ You mean F(p)<1 for p<1/2 - there is always a positive probability of the starting square being boxed in. $\endgroup$ Jan 18, 2023 at 13:29
  • 1
    $\begingroup$ Er, sorry, yes. I was mixing up the probability that a given place (say the origin) is part of an infinite component, with the probability that there is an infinite component. I don't know whether the precise question asked here is one whose answer is fully known. $\endgroup$
    – Gareth McCaughan
    Jan 18, 2023 at 15:45
  • 1
    $\begingroup$ @bobble I came up with the question myself when I was thinking about the backrooms but maybe someone else has thought of it before me. I am not sure. $\endgroup$ Jan 19, 2023 at 5:40

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.