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?

  • $\begingroup$ 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
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    $\begingroup$ 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
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    $\begingroup$ 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
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    $\begingroup$ 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
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    $\begingroup$ @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


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