On Spatial Matchings: The First-in-First-Match case

In this paper, we describe a process where two types of particles, marked by the colors red and blue, arrive in a domain $D$ at a constant rate and are to be matched to each other according to the following scheme. At the time of arrival of a particle, if there are particles of opposite color in the system within a distance one from the new particle, then, among these particles, it matches to the one that had arrived the earliest. In this case, both the matched particles are removed from the system. Otherwise, if there are no particles within a distance one at the time of the arrival, the particle gets added to the systems and stays there until it matches with another point later. Additionally, a particle may depart from the system on its own at a constant rate, $\mu>0$, due to a loss of patience. We study this process both when $D$ is a compact metric space and when it is a Euclidean domain, $\mathbb{R}^d$, $d\geq 1$. When $D$ is compact, we give a product form characterization of the steady state probability distribution of the process. We also prove an FKG type inequality, which establishes certain clustering properties of the red and blue particles in the steady state. When $D$ is the whole Euclidean space, we use the time ergodicity of the construction scheme to prove the existence of a stationary regime.