Scaling Limit of a Stochastic Clustering Model on $\mathbb{R}$

We consider an infinite-dimensional stochastic clustering model on $\mathbb{R}$. In discrete time, each point of a unit-intensity simple point process moves halfway toward either of its left or right neighbors, chosen uniformly at random. Co-located points are merged into a single point, and the resulting simple point process is rescaled to unit intensity. We show that, when the point processes are shifted so that there is a point at the origin, the dynamics have a unique weak limit when the initial point process is renewal. For this limiting point process, the gap distribution has exponential tails. We also show that for the time-reversed process and with an appropriate scaling in space, there is a limiting (random) distribution function on $\mathbb{R}$, whose associated measure assigns to $\mathbb{R}$ a measure corresponding to the gap between consecutive points. Finally, we discuss several relevant research directions.