I hope my title asks the question reasonably well, but I'll add some rigor here. Consider a process as follows: at step $ n = 0 $ we start with a set consisting of one point, $ S_0 = \{(0,0)\} $. For each $ n \geq 1 $ we compute $ S_n $ as follows. Let $ T_{n-1} $ denote the "lattice boundary" of $ S_{n-1} $, i.e., the set of lattice points which are "adjacent" to points in $ S_{n-1} $ but not in $ S_{n-1} $. For example $ T_0 = \{(1,0), (0,1), (-1,0), (0,-1)\} $. Now $ S_n $ is constructed by adding adding exactly one uniformly-randomly-chosen point from $ T_{n-1} $ to $ S_{n-1} $.
I thought of this today while playing Carcassonne, in which the game board mimics this process (kind of). The board seemed to be roughly a ball at the end.
Conjecture: For large $ n $, $ S_n $ is almost-certainly nearly (the set of lattice points inside) the Euclidean ball of volume $ n $ centered about the origin.
Stronger (and more dubious) Conjecture: An analogous statement is true if we run an analogous process in the $ m $-dimensional lattice $ \mathbb{Z}^m $ for any dimension $ m $.
I believe the use of "almost certainly nearly" is going to be understood by most readers, but I can elaborate on a definition with some epsilons and deltas if too many readers are head-scratching.
Question for readers: Do you have proofs or ideas for proofs or ideas why either conjecture might be false? I am surprised this question was not easily found by my Google search or stack exchange search. It seems quite natural.
My thoughts My mathematician friend thought that the "large surface area phenomenon" in high dimensions might give rise to "highly branching" non-ball-like shapes in high dimensions, but we agreed it's probably true when $ m = 2 $. If I had to bet, I'd say the stronger conjecture is also true though.
A picture for $ m = 2 $, $ n = 10,000 $ with superimposed sphere of radius $ \sqrt{10000/\pi} $
$ m = 2 $, $ n = 10,000 $ with superimposed sphere of radius $ \sqrt{10000/\pi} $" />