I am wondering if this random walk remains finite with positive probability. Start with three lines $A,B,C$ that are extensions of an equilateral triangle. Let $p_0$ be one corner. Generate a line $L_1$ through $p_0$ at a random orientation. Now, from $p_0$ walk either right or left with equal probability along $L_1$ until you hit the first intersection point $p_1$ of $L_1$ with $\{A,B,C\}$, or reach $\infty$. Clearly there is a $\frac{1}{2}$ chance you stay finite. Through $p_1$ generate a randomly oriented line $L_2$, and walk right or left to the first intersection point $p_2$ of $L_2$ with $\{A,B,C,L_1\}$. And so on.

On the one hand, there are always routes to $\infty$. On the other hand, as the line arrangement thickens, the number of steps to escape grows. So if the process survives a few iterations, it becomes less likely it will escape. Intuition is likely useless here, but it feels to me like this should remain finite. Can anyone see an argument?

If it does remain finite, then other questions suggest themselves, but perhaps I should start with the basic infinite/finite question.

Here is a hand-executed example (corrected from the original by Pablo):

Update. Here is a first attempt at an implementation:

I just stopped the iteration after $p_8$. I'll report more substantively after I have collected some data.

Addendum. Great progress! Bill Thurston's convincing analysis coupled with independent simulations. I appreciate the interest! Of course many interesting questions remain: What is the probability of remaining finite? What is the distribution of the number of steps before shooting off to $\infty$? When a path remains finite forever, does its total length remain bounded?

5 Added one image from an implementation.

I am wondering if this random walk remains finite with positive probability. Start with three lines $A,B,C$ that are extensions of an equilateral triangle. Let $p_0$ be one corner. Generate a line $L_1$ through $p_0$ at a random orientation. Now, from $p_0$ walk either right or left with equal probability along $L_1$ until you hit the first intersection point $p_1$ of $L_1$ with $\{A,B,C\}$, or reach $\infty$. Clearly there is a $\frac{1}{2}$ chance you stay finite. Through $p_1$ generate a randomly oriented line $L_2$, and walk right or left to the first intersection point $p_2$ of $L_2$ with $\{A,B,C,L_1\}$. And so on.

On the one hand, there are always routes to $\infty$. On the other hand, as the line arrangement thickens, the number of steps to escape grows. So if the process survives a few iterations, it becomes less likely it will escape. Intuition is likely useless here, but it feels to me like this should remain finite. Can anyone see an argument?

If it does remain finite, then other questions suggest themselves, but perhaps I should start with the basic infinite/finite question.

Correction to figure

Here is a hand-executed example (thanks to Pablo!)corrected from the original by Pablo): $p_1=L_1 \cap A$, $p_2=L_2 \cap B$, $p_3=L_3 \cap C$, $p_4=L_4 \cap A$

Update. It Here is a first attempt at an implementation:

I just happenstance that with stopped the random numbers I used, all points (so far) are on $A$, $B$, or iteration after $C$.p_8\$. I'll report more substantively after I have collected some data.

4 edited tags
3 Corrected figure.; added 9 characters in body
2 Alex R.'s comment
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