How can I calculate the limit of a transition matrix?

Question

I've been wondering about how can I calculate the limit of this matrix: These states the different movements of the knight within a $4\times3$ chessboard, and what I'm trying to do with the limit is to get an insight of the long-term probability of being on each and every square, in order to analyze which squares are the most difficult to get to or the once less likely to be visited on a random knight's tour attempt. I really appreciate the help.

Answer 1

Given that the markov chain is irreducible and aperiodic, which it is, the limiting distribution coincides with the unique stationary distribution, which can be found as the solution $\pi$ to the equations: \begin{equation} \pi X = \pi \quad \text{ and } \quad \sum_i \pi_i = 1 \tag{1}\end{equation} where $X$ is the transition matrix. You are asking specifically for the limit of the transition matrix $X^n$, but since the limiting behaviour cannot depend on the initial distribution, we must have that all rows are equal in the limit. In fact all rows of the limiting matrix will be equal to the limiting distribution $\pi$, which can be found as the solution to $(1).$