Consider the **torus** graph, or the **toroidal grid**, which looks like

(The graph's vertices are the bold dots).

I will discuss only square torus graphs, where there is an equal number of vertices in a "vertical" circle and a "horizontal" circle. Such a graph can be represented by a cyclic square matrix, where every entry is a vertex.

Take every vertex and assign it a random value - 0 or 1 with equal probability. The matrix then looks like

Similar to other questions I asked, start the following process: at each step, pick a random vertex, and update its value to the value of most of its neighbors (in case of draw the value remains unchanged). It is known, that the process always converges, i.e. each vertex changes value only a finite amount of times.

Take a look:

Another example, slower one:

I found that the distibution of the final amount of 1's, as well as the distribution of the ratio between the final amount of 1's and the initial amount of 1's, both look like normal, when the torus dimensions are constant, and each iteration is a new random choice of initial conditions.

Final amount of 1's (matrix size is 30x30):

Ratio distribution:

I'm looking for a way to prove that these distributions converge to normal when the torus size (which is the size of the matrix) goes to infinity, and if they really do, find how the standard deviation depends on the torus size.

Thank you!

**Edit:** A possible direction could be Stein's method, although I don't know how to implement it here.

**Edit 2:** Let $X$ be the final proportion of 1's. $E[X] = \frac{1}{2}$, because if $Y$ is the final proportion of 0's, then $E[X] = E[Y]$ and $X + Y = 1$. If you manage to calculate $Var[X]$, this will be a good start!

**Edit 3:** Perhaps a variant of the central limit theorem for dependent variables is needed.