Assume a finite 2-dimensional grid as shown in the attached figure. In the grid there are two ”anchor” points, a ”red” and a ”black” represented by the points $r$ and $b$. The distance of r and b is $d$. A ”painter” moves randomly in the grid. Once it reaches one of the two anchors it paints its brush with the corresponding color of the anchor and paints every node of the grid it subsequently visits. Alas, the amount of paint in the brush can last only for $l$ nodes to be painted. What is the probability a grid point is red in the long run?
I have some obvious results for the 1D case with $l=\infty$.
I also have a cool Python program to visualize the system.
If you are interested in the more general case, let me know.