One-Dimensional Random Walk with Phase Transition

We consider a random walk on a one-dimensional lattice. The walk is asymmetric but with different asymmetry on the right and left halves of the line. As the parameter space describing the two asymmetries is covered, several qualitatively different distributions result: limiting distribution, unimodal diffusion and bimodal diffusion. The corresponding parameter space phase boundaries are obtained, as well as the precise form of the distributions.