Toeplitz matrices and random walks with memory

We use a technique based on Toeplitz matrices to calculate the probability distribution for certain random walks on a lattice in continuous time where the walker can take steps of various sizes in each direction and where the probability of a step depends on the nature of a finite set of previous steps. If k(ij) is the rate constant for a step of j units given a history of type i, then we can solve the random walk problem for the special case when the sum over k(ij) is independent of j.