Deterministic walk on Poisson point process

Abstract

A deterministic walk on a Poisson point process in Rd is an oriented graph where each point of the process is connected to only one other point following a deterministic and stationary rule of connection. In the paper we investigate the absence of percolation for such graphs and our main result is based on two assumptions. The Loop assumption ensures that any forward branch of the graph merges on a loop provided that the Poisson point process is augmented with a finite collection of well-chosen points. The Shield assumption ensures that the graph is locally determined with possible random horizons. Among the models which satisfy these general assumptions and inherit in consequence the finite cluster property, we focus on the deterministic walk to the k-th neighbour, with k any integer greater than one.