TITLE: Limit shape for fixed-perimeter lattice polygons
QUESTION [11 upvotes]: 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:

Q. 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!

Animation of 200 operations, starting from a crenellated square. The green dot is the centroid of the vertices.

REPLY [2 votes]: Sometimes models like this can simplified e.g. loop erased random walk and self-avoiding walk.  Your model may even be integrable if you allow for a small amount of self-intersections.

There are certainly lots of 2D growth models (e.g. Laplacian growth) and they all tend to fall into the KPZ universality class.  However, we ought to prove that your model also falls in this class.  And maybe there already exists theorem which accomplishes as much. The notes certainly say "ballistic deposition" (as of 2012) is unknown.  [1]
It may be helpful to use a small amount of topology here.  Your curve is always contractible so that 
$$ \gamma \equiv 0 \in H_0(\mathbb{R}^2) $$
and I'm phrasing it in this rather pretentious way in order emphasize that your model is equivalent to adding squares at random from the interior inside the boundary or vice versa.  
Homology is clearly not refined enough to detect curvature (in the "hydrodynamic" limit).  I already suspect there can be no limit shape at all.  If I start with a perfect circle, we're gently adding bits of curvature everywhere, there's no reason why these should perfectly cancel out globally.