Eigenfunction approach to the persistent random walk in two dimensions

The Fourier–Bessel expansion of a function on a circular disc yields a simple series representation for the end-to-end probability distribution function w(R,φ) encountered in a planar persistent random walk, where the direction taken in a step depends on the relative orientation towards the preceding step. For all but the shortest walks, the proposed method provides a rapidly converging, numerically stable algorithm that is particularly useful for the precise study of intermediate-size chains that have not yet approached the diffusion limit.a