I am familiar with discrete markov processes and idea of their convergence towards a stable distribution after some number of steps. But my question is much simpler--more like a validation of some intuition.
Say I am looking at the transition of population between 3 cities. I have a $3\times3$ transition matrix $M$ with entries $m_{ij}$ for the transition probability for a person moving from city $i$ to city $j$. I also have a vector $v_0$ of initial values--in this case the initial populations of the 3 cities.
My question is how to determine the population of the 3 cities after $k$ transitions? That is, we would like to know $v_k$.
Seems like the simplest results is akin to what we would see in a simple differential equation. Something like:
$$ \text{population after first transition} = v_1 = v_0M $$
The population after a second transition would be:
$$ v_2 = v_1M = (v_0M)M $$
If we iterate this forward, then we would get:
$$ v_k = v_0M^k $$
This seems simple enough, but just wanted to make sure I did not make some careless error or leave something out. Of course I can do an eigenvector decomposition of $M^k$ to quickly determine the matrix power for something larger than a $3\times3$ matrix.
You appear to be confusing two different, but related processes. The $3\times3$ matrix $M$ gives the transition probabilities for an individual person moving from city to city. As you follow this one person around, the matrix $M^k$ then gives the $k$-step transition probabilities, that is, the probability that that particular person ends up in city $j$ after $k$ steps if she started in city $i$.
For the aggregate populations, on the other hand, the system is much larger. Depending on how you model it, you might have $N^3$ states, where $N$ is the total population (in this model you track the location of each individual), or $\binom{N+2}{2}$ states (in this model you track the population of each city, with a total population of $N$). Either way, the transition matrix is a lot larger than $3\times3$. Now, if you want to examine the expected population distributions, you can use your individual $3\times3$ matrix: by linearity of expectation, if you have a three-element vector with the city populations, then multiplying it by $M^k$ gives you the expected population distribution, not the actual one, after $k$ steps.