Simple random walk on \(\mathbb{Z}^2\) perturbed on the axes (renewal case) (Q6177519)

The authors study a simple random walk on \(\mathbb{Z}^2\) with constraints on the axes. The model under consideration is partly inspired by an experiment in physics when a quasi-two dimensional cloud of cold neutral atoms is submitted to a pressure forces exerted by a laser beam. The authors simplify the model in many directions, namely, they consider a single particle instead of a system of particles (the gas of atoms), and take a simple form for the applied force which decreases when the particle is getting closer to the origin. This simulates that at a certain moment the pressure exerted by the force is not enough to confine the particle in a single point. It is assumed that a particle evolves freely in the cones but when touching the axes a force pushes it back progressively to the origin. The main result proves that this force can be parametrized in such a way that a renewal structure appears in the trajectory of the random walk. This implies the existence of an ergodic result for the parts of the trajectory restricted to the axes. The main ideas of the proof of this result are carefully explained, which makes the paper convenient for the reader.