I have a random walk on a chessboard, where each square is a state/vertex.
I understand that the location of pieces on the board describes a Markov chain, and one way to look at this is through a random walk. So for instance, on a standard $8 \times 8$ chessboard, the movement of a Knight is a random walk.
And the Markov chain described by the Knight’s movement is irreducible. This is because starting from any square/state, the Knight can reach all other squares/states on the $8 \times 8$ board.
We can also find the stationary distribution for any vertex/square by considering the Knight’s random walk on this connected graph. This will be:
Stationary distribution at vertex $(v)$ $=$ degree $(v)$ / sum of the degrees of all vertices,
where the degree of $v$ is the number of vertices/squares the Knight can reach in one step, starting from vertex/square $v$.
Now, instead of a finite state space, suppose I have an infinite state space. So, I am imagining the chessboard to not have any corners or edges, and the board keeps extending indefinitely in all directions.
I am trying to understand the behavior of the Markov chain described by the Knight’s movement here. We have a random walk in $2$ dimensions with infinitely many states. And the edges are L-shaped moves of the Knight that connect successive vertices/squares.
Somewhat related to this set up is the result that a simple random walk on $Z^2$ is recurrent. Now, if we know that the Knight starts from a certain state/square in the infinite chessboard, can we say that the Knight will certainly return to the starting square with probability $1$? But my issue is that the random walk given by the Knight’s movement is different from a simple random walk on $Z^2$.
Given this, how can we prove that the Knight’s movement is also recurrent in an infinite chessboard? As I understand, the Knight’s movement is null recurrent, but I would like to prove this. Any insights would be most helpful.