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

TL;DR

This paper revisits the Gillis model, a non-homogeneous random walk with position-dependent drift, demonstrating its role as a discrete analogue for diffusion in a logarithmic potential, and presenting both classical and new findings.

Contribution

It provides a comprehensive analysis of the Gillis model, highlighting its significance as a discrete representation of diffusion in logarithmic potentials with exact results.

Findings

Gillis model allows for exact analytical results.

It serves as a discrete analogue for diffusion in logarithmic potentials.

The paper presents both classical and novel insights into the model.

Abstract

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