The dynamical equation for ${\cal P}_i^{(n)}$ can be written directly from the description above. The probability of being in the state $i$ at step $n$ is given by the probability of being in any state $j$ at the previous step, and making an accepted transition to $i$ added to the probability of being in the state $i$, making a transition to any state $j$ and rejecting the move: $${\cal P}^{(n)}_i = \sum_j \left [ {\cal P}^{(n-1)}_jT_{j\rightarrow i} A_{j\rightarrow i} +{\cal P}^{(n-1)}_iT_{i\rightarrow j}\left ( 1- A_{i\rightarrow j} \right) \right ] \,.$$ Since the probability of making some transition must be 1, $\sum_j T_{i\rightarrow j} = 1$, and the above equation becomes $${\cal P}^{(n)}_i = {\cal P}^{(n-1)}_i + \sum_j \left [ {\cal P}^{(n-1)}_jT_{j\rightarrow i} A_{j\rightarrow i} -{\cal P}^{(n-1)}_iT_{i\rightarrow j}A_{i\rightarrow j} \right ] \,.$$