I have a random walk on $\mathbb{Z}^2$ that takes a step
with equal probability in the three directions that avoid
retracing the previous step.
The walk proceeds until it returns to a lattice point
previously visited, at which time it pinches off a simple,
closed loop or polygon:

I'd like to know the distribution of the perimeters of
these polygons.
(In the example above, the perimeter is 10.)
Simulations show that the average perimeter is about 5.6,
with perimeter 4 the overwhelming favorite,
as one would expect:

I feel this distribution must be known to the experts and
not difficult to explicitly detail, but
after looking at hitting times, first-passage times, self-avoidance times,
and various other frequently studied random walk quanities,
I am not finding a close-enough analog to help.
Thanks for any pointers you might provide!

**Addendum.**
Here is log-plot of the probability of a perimeter of length $L$, based on a simulation of $10^6$ walks. The first point represents 642,225 perimeters for $L=4$, the second point 176,043 perimeters for $L=6$, etc. The last point plotted is 135 instances of $L=38$. (There is one polygon of length $L=74$ in these million trials.) The average perimeter length is 5.62, which occurs after an average of 8.46 steps.

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