Let $P$ be a simple polygon defined by $n$ unit-length segments
connecting lattice points of $\mathbb{Z}^2$.
I have two operations that preserve the perimeter of $P$.
The first is the "pop" of a corner either inward or outward;
the second is pushing in a 3-edge "tab" at one spot and immediately pushing out
a tab at another. Both operations are only applied if
they preserve simplicity (non-self-intersection):

My question is:

What is the limit shape of $P$ as $n \rightarrow \infty$ and the operations are applied randomly for sufficiently many iterations?

I realize this question is imprecise. Perhaps the limit shape depends on the frequency balance between the two operations, which I have not specified. In addition, I am not quite sure how to best to describe a shape. But even an intuitive answer would be useful: Will the shape tend toward a convex-like blob, or toward a more spidery shape? (My guess is the latter.)

I was hoping to get a sense with small experiments as below, but it may require much larger $n$ and many more iterations for a clear pattern to emerge. I would appreciate any help in sharpening the question, answering it in some form, or pointers to related literature that could shed light on the issue. Thanks!

_{(The animation shows 800 operations. The green dot executing a random walk is the centroid of the vertices. The starting configuration is a crenellated square.)}