First passage times for (persistent) random walkers: a mystery

A random walker hops between sites of a lattice, choosing a new direction at each hop. A persistent random walker has a memory of its hopping direction, so the same direction may be used for multiple hops. In both cases we can ask how long it takes for a particle to fall off the edge of finite lattice: this defines a first passage problem. We find that the mean first-passage time, starting from an edge of the lattice, is the same for a random walker and a persistent walker. This implies that rate at which a persistent walker changes direction does not enter into this first-passage time. The reason why is (at present) a mystery; with perhaps a walker that interpolates between these two behaviours shedding some light (or not) on it...