Counting oriented rectangles and the propagation of waves

We propose a simple counting problem involving chains of rectangles on a planar lattice. The boundaries of the chains form a type of random walk with a finite inner scale. With orientation neglected, the continuum limit of the walk densities obeys the Telegraph equation, a form of diffusion equation with a finite signal velocity. Taking into account the orientation of the rectangles, the same continuum limit yields the Dirac equation. This provides an interesting context in which the Dirac equation is phenomenological rather than fundamental.