The probability that a random walk on $\mathbb{Z}^3$ returns to the origin is about 34%.
This is (part of)
Pólya's theorem.
I have been looking for an analogous (numerical) result for the probability
that a random "myopic" self-avoiding walk on $\mathbb{Z}^3$ returns to the origin to form a self-avoiding polygon.
By a "*myopic* self-avoiding walk," I mean
a walker who takes a sequence of random (unit) steps, each restricted to avoid previously visited sites (to quote Yoav Kallus's and Vincent Beffara's comments).
Any other related information—e.g., probability of the walk
ending in a cul-de-sac before $n$ steps (see below)—would be welcomed,
as would either small-$n$ results or asymptotics.

^{Origin: green. Cul-de-sac: red, reached after 335 steps.}

A 2D version of this question was posed earlier, but not entirely answered: "Probability that a “closable” self-avoiding random walk forms a polygon"