Let $\mathcal {G}$ be an $n\times n$ grid inhabited by $k$ animals, time is indexed by the naturals, the metric $d(x_i,x_j)$ on $\mathcal G$ between two animals $i$ and $j$ is the Manhattan distance. Each animal obeys the following rules:
- It always inhabits a specific square in $\mathcal G$
- It can move only one square at a time within its current Moore neighborhood and does so randomly.
- No two animals can inhabit the same square at the same time.
- If $2<d(x_i,x_j)\le 4$ animals $i$ and $j$ are twice as likely to move toward each other as away from each other.
- If $d(x_i,x_j)\le 2$ animals $i$ and $j$ are twice as likely to move away from each other as toward each other.
Let $X_m(t) = 1$ iff there is an animal on the $m$th square at time $t$, and $0$ otherwise. Given an initial configuration of the $k$ animals on $\mathcal G$, is there a way to calculate $\mathbb{P}(X_m(t)=1)$?
I have coded a simulation on my computer, although my computer can hardly keep up with $k=47$ and $n=100$, these were the highest stats I could get. I know that getting an exact answer to the problem will probably be very complicated, but I was hoping for estimations or even just bounds on the probability. I would be content with learning new ways of looking at this problem and hearing of new techniques.
I have tried markov chains but the fact that animals are able to change their behavior based on the behavior of other animals makes it difficult to find the values of the entries for the matrices. Any help would be appreciated.