The zig-zag walk with scattering and absorption on the real half line and in a lattice model

The Darwin–Hamilton equations, describing one-dimensional transport with scattering and absorption, are expanded into a recursion. The solution involves ballot numbers. The recurrence probability as function of scattering order is given by Catalan numbers. To reproduce this analytical result in a lattice model, a novel relation between Narayana and Catalan numbers is derived. (paper)