Suppose that we have a chess and a knight starts from bottom-left.Each time makes one of the allowable knight's moves, what's the expected time that takes to return at the bottom-left.
My problem is to find the transition matrix.
It will be a $8\times 8$ matrix,that each row will sum to 1.
My approach is that if we start from square (8,8) (bottom-left) then there are two possible moves , (6,2) or (7,3) with probability $\frac{1}{2}$ each.At the next turn we will have the following allowable moves (4,1),(4,3),(5,4),(7,4) if we continue from square (6,2) and if we continue from square (7,3) we will have (5,1),(4,2),(4,4),(5,5),(7,1),(7,5).So we conclude that , let's say for 4 , that there is a path that connects 4 with (1,2,3,4,5) but if we continue with more setps we will see that 4 is also connected with (6,7,8).So I think that the transition matrix will be $8\times 8$ with probability entries $\frac{1}{8}$.
Is my approach correct ?? Sorry in advance if my text is confusing.