Consider a two dimensional square lattice ($n$ by $n$), which is our space $S$ (each point labelled by an index $1\to n^2$), containing two types of particles, distinguished here by either an index $1$ or $2$. There are $4$ of them on the lattice, $A_1$, $A_2$, $B_1$ and $B_2$. (Note that $A$'s are distinguishable from $B$'s).

The system evolves stochastically for a span of time $t$, where each particle is allowed to make a random jump at each step (i.e. $1$ particle jumping per step), within the lattice. The type $1$ particles have a different discrete jump compared to type $2$ ones, namely for the former the jumps can only be diagonal (i.e. both coordinates change by same amount $\Delta_1=(\pm \alpha,\pm \alpha)$) and for the latter they are lateral (i.e. only one coordinate can change for each jump, $\Delta_2=(\pm \beta, 0)$ or $(0, \pm \beta)$). No two particles can occupy the same point.

- Given that we have 4 particles, the states are given by vectors of $n^2$ components (Initial state: $I_0=(A_1, A_2, B_1, B_2, \cdots)$) and $T$ is an $n$ by $n$ matrix. A specific example one could consider $n=4,$ $\alpha=1$ and $\beta=2$ (if the general case is difficult to discuss about).

Given only the above (type of lattice, particles and types, size ...), what extent of the transition matrix $T$ of this random walk of 4-particles can already be written down in order to study its emergent properties?