# Grid and Probability

Consider the first quadrant of a $2D$ Cartesian system as a grid. A recursive function "marks" solely the boxes in the grid (one at a time) that neighbor exactly two other marked ones. Such boxes are referred to as "kink sites". The x and y axes are assumed to be already marked as an initial condition. At each iteration, all kink sites have equal probability of getting marked. So, e.g. starting with the unmarked grid the first and only box that can be marked is the one adjacent to the corner of the grid (box $(1,1)$). Subsequently, there are two options; either box $(1,2)$ above, or box $(2,1)$ to the right of the marked corner box, both equally likely to be marked with probability $0.5$. The following kink sites keep getting marked in the same manner. The attached image (grid not shown) shows all marked boxes after $10,000$ computer simulated iterations. Different colors represent any bunch of $1,000$ iterations. As the number of marked boxes grows infinitely, if any, what algebraic function is traced by the kink site front?

Note by Tomasz Bukowy
1 month ago

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