Consider a random walk on an (m+1) × (m+1) triangular grid, illustrated below for m = 6.

The points of the grid are labeled . From the point (j,i), a transition may take place to one of the four adjacent points (j+1,i), (j,i+1), (j-1,i), (j,i-1). The probability of jumping to either of the nodes (j-1,i) or (j,i-1) is

with the probability being split equally between the two nodes when both nodes are on the grid. The probability of jumping to either of the nodes (j+1,i) or (j,i+1) is

with the probability again being split when both nodes are on the grid. If the (m+1)(m+2)/2 nodes (j,i) are numbered in some fashion, then the random walk can be expressed as a finite Markov chain with transition matrix A of order consisting of the probabilities a_{kl} of jumping from node l to node k (A is actually the transpose of the usual transition matrix; see [Feller]).

We are primarily interested in the steady state probabilities of the chain, which is ordinarily the appropriately scaled eigenvector corresponding to the eigenvalue unity.

Figure 1 shows the sparsity pattern of the resulted random walk matrix of order 136 (i.e. m=15).

To calculate the i-th element of the vector Ax one need only regard the components of q as the average number of individuals at the nodes of the grid and use the probabilities (5) and (6) to calculate how many individuals will be at node i after the next transition.