As per Ron Maimon's suggestion, I did a little simulation for the lattice version, counting the number of paths
that get trapped at or before $n$ steps.
For example, here is a trapped path of 45 steps:
Because there is a positive probability at any point of forming a shape like the letter 'G'
(if there is sufficient room), the probability of trapping goes to 1 as $n \rightarrow \infty$.
For $n=100$, the probability is already over $\frac{3}{4}$.
Addendum. Incidentally, I have some evidence—not definitive—that the mean path length before
reaching a cul-de-sac is about 71.6 steps.
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2 | Addendum on mean path length. | ||
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As per Ron Maimon's suggestion, I did a little simulation for the lattice version, counting the number of paths
that get trapped at or before $n$ steps.
For example, here is a trapped path of 45 steps:
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