Let us consider a classic $8\times 8$ chessboard (grid), where a lone rook is doing a random walk. We assume that the rook starts its journey from the $(7, 0)$ square ($A1$), and that there are no other pieces on the board that could "block" the rook's movements.
We know that the rook's graph is a perfect graph, and thus the number of available moves at any square of the chessboard for the rook will be 14. I want to infer the stationary distribution of the Markov chain that corresponds to the rook's walk.
My initial assumption was that the distribution must be close to a two-dimensional discrete uniform distribution (something like $U_2\{0, 7\}$), i.e. the limiting probability for the rook to occur at any square of the board will be about $1/64$. I tried doing a simulation for the process in python, and what I've obtained did not really match my expectations. The following image represents the stationary distribution that results from the simulation (the color scheme is "viridis", the brighter the color is, the higher the probability. The sum of all limiting probabilities is $1$):
For some reason it doesn't seem to follow the uniformness that I was expecting. The code that I wrote for sure does work as intended, since I've also tested it on deriving the stationary distribution for the knight's random walk, and I got the very same result as in here. So the issue seems to not be code-related.
If anyone helps me to identify the issue in my logic and probably to navigate me in constructing the stationary distribution for the rook's MC, I'll be very grateful!
Thank you in advance!
P.S. One can find the code that I wrote here.