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It feels as though this problem should be well-known, so I'm happy to simply be provided a reference if this is the case.

Consider a modified random walk in $\mathbb Z^2$ that proceeds as follows:

  1. The random walk starts at the origin.
  2. At each point $(x,y)$, we consider the set $S$ given by the points $\{ (x+1, y), (x-1, y), (x, y+1), (x, y-1) \}$ that have not yet been visited, and choose one at random to walk to. If at any point, this set $S$ is empty, the walk terminates.

In this way, the walk visits each point at most once, and terminates if at some point each of the four adjacent points have already been visited. For instance, here is such a walk I simulated which starts at the origin in blue and terminates at the point in red.

Random walk example

Questions:

  • It seems as though the walk always terminates. Is it possible to prove that this occurs with probability 1?
  • Is anything known about the distribution of the number of steps taken before termination?
Randall
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    What exactly do you mean by the set being empty? Are you considerung a sublattice of $\mathbb{Z}^2$? When does your walk terminate? – Anton Steinfadt Jan 27 '24 at 17:47
  • @AntonSteinfadt Edited to clarify. – Randall Jan 27 '24 at 17:53
  • Concerning your first question, you may use the (even stronger) result that a simple two dimensional random walk is recurrent (Polya's theorem). I know that the number of steps until the random walk self-intersects the first time have been studied. (see for example: https://mathoverflow.net/questions/325052/how-many-random-walk-steps-until-the-path-self-intersects) Your random walk is a little different since it does not "give up" until there is really no possibility left. – Anton Steinfadt Jan 27 '24 at 18:17

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Given that I interpret your description right, you're basically simulating paths of a 'self-avoiding random walk' (SAW). Concerning your second question, you should refer to this question. Gabriel Romon cites this document which deals with your question.

Anton Steinfadt
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