Exploring the Gillis model: a discrete approach to diffusion in logarithmic potentials

The Gillis model, introduced more than 60 years ago, is a nonhomogeneous random walk with a position-dependent drift. Though parsimoniously cited both in physical and mathematical literature, it provides one of the very few examples of a stochastic system allowing for a number of exact results, although lacking translational invariance. We present old and novel results for this model, which moreover we show represents a discrete version of a diffusive particle in the presence of a logarithmic potential.