Consider the symmetric random walk on $\mathbb{Z}^2$, where you go in one of the four directions with probability 1/4. We start in 0. My question is whether there are results on counting how many ways there are to do a random walk of $n$ steps and get back to the starting point without ever crossing a point where you have already been, except of course the starting point. Thank you very much.
2026-04-03 00:37:28.1775176648
number of loops of length $n$ without crossings in random walk on $\mathbb{Z}^2$
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I believe you want sequence A010566 at OEIS; there are many references. There are many related sequences as well; the general search term you want is self-avoiding [closed] random walk.