I know that the rate of convergence of the power method is given by $|λ_2/λ_1 |^k→0$. Does that imply the fastest convergence happens when $λ_2 = ... = λ_n = 0$? That is, the characteristic polynomial of the Markov chain is $(x-1)x^{n-1}$ or it has exactly n-1 zero eigenvalues. If you look at the Jordan canonical form of such a matrix, the power method will converge in at most n-1 iterations, as the off-diagonal ones will go away as you raise the power (actually, it will converge in k iterations, where k is the size of largest eigenblock along the diagonal).
Edit: I think the word "fastest" I used initially might not be the best choice of vocabulary, as I myself have trouble to find the precise meaning of it. While it's true that I was asking for some sort of convergence interpretation when $λ_2=0$ given the fact that the convergence rate is $|λ_2/λ_1|$. Does the error in this case mean anything at all since you will have an exact solution at the end of your $(n-1)$-th iteration? Feel free to elaborate any further.