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

TL;DR

The paper analyzes a continuous-space, two-color FIFM matching process where red and blue particles arrive at rate $\lambda$ and match within distance $1$ to the earliest opposite-color particle, with independent impatience rate $\mu$. On compact domains, it proves a product-form stationary distribution via time-reversal and local balance, and establishes an FKG lattice property implying clustering of same-color particles. For the Euclidean setting, it constructs stationary regimes through local coupling and a Coupling from the Past framework, proving existence and describing coupling-time behavior. The results connect continuum matching dynamics to Widom-Rowlinson-type processes, exposing positive associations and potential phase-transition-like clustering, with avenues for future work on infinite-volume limits and symmetry-breaking phenomena.

Abstract

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, $μ>0$, 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.

Figures (1)

• Figure 1: An illustration of the FIFM spatial matching process. The vertical dimension represents the set $D$. The rectangles represent the lifetimes of particles in the system -- so, the vertical dimension of the rectangle represents the spatial range of interaction of a particle, a solid disk to the left of a rectangle represent its arrival, and a hollow circle at the right represents its departure. The set of particles present in $\eta_t$ are marked by crosses; these are those particles whose rectangles intersect the "vertical line" at time $t$.

Theorems & Definitions (35)

• Lemma 2.1
• proof
• Definition 2.2: Definition of a valid state of $\hat{\eta}_t$
• Theorem 2.3
• Corollary 2.4
• proof
• Remark 2.5
• Lemma 2.6
• Theorem 2.7
• proof