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It's well-known that a random walk in two dimensions almost always returns to the starting point; equivalently it almost always visits every two-dim location. And in three dimensions, this is not true.

My question is whether the same holds true for two-dimensional infinite mazes. That is, consider the set of locations to be the two-dimensionalprobab integer lattice points with some, but not all, adjacent pairs of points joined by doors. Thus, this graph is a maze. If two points are connected in the graph, will a random walk starting at one point almost always visit the other point?

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  • $\begingroup$ Surely you can have infinitely many pockets where a lattice point is blocked from all four adjacent points? This would mean you don't almost always visit other points (there are infinitely many exceptions, i.e. the squares inside a pocket). $\endgroup$ Commented 13 hours ago
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    $\begingroup$ There's guaranteed recurrence only in the connected component of the maze that the walker starts in $\endgroup$ Commented 13 hours ago
  • $\begingroup$ Some references on the general problem can be found here: mathoverflow.net/questions/284113/… $\endgroup$ Commented 12 hours ago
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    $\begingroup$ @Amitai The question says "If two points are connected in the graph..." $\endgroup$ Commented 4 hours ago

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