I believe the following problem is related to something called the "voter model" in statistics. This is not my area of expertise so please forgive me if the answers turn out to be well known.
Consider a vector $V$ of length $n$ where the elements are integers chosen from $[m]$. At each (discrete) time step we apply the following update rule to $V$.
- With probability $p$, choose an index $i$ uniformly at random from $[n]$ and set $V_i = x$ where $x$ is uniformly chosen from $[m]$.
- With probability $1-p$ choose two indices $i$ and $j$ uniformly at random and set $V_i = V_j$. That is copy the value $V_j$ to replace the value $V_i$.
When $p$ is reasonably small (but not too small) the vector spends most of its time as a (more or less) constant vector with all elements the same except for a small number of single indices which change and than change back. It also periodically transitions rapidly to be a different (more or less) constant vector with the same properties.
I am interested in understanding this process in more detail. For example, how long does it spend in each (more or less) constant state (i.e how long until it flips to another one), how long does it spend not in a (more or less) constant state?