A particle system with mean-field interaction: Large-scale limit of stationary distributions

We consider a system consisting of n particles, moving forward on the real line. The particles move in jumps. The system state at a given time is the current empirical distribution of particle locations. Each particle gets "urges to jump" as an independent Poisson process of constant rate. However, a particle getting a jump urge actually jumps with the probability given by a decreasing function of the particle's location quantile within the current state (i.e., empirical distribution); hence this a mean-field type of particles' interaction with each other. When a particle does jump, the jump size is independent, distributed as a random variable Z > 0. We are interested in the system behavior when n is large. This model was introduced in [5,6] as an idealized model of distributed parallel simulation. In this case n particles represent n processors("sites") simulating different part of some large system, and a particle location is the current "local simulation time" of the corresponding processor. The following types of questions are of interest, as n becomes large: how the local times of the processors progress over time; do local times "stay closely together;" does the evolution of the empirical distribution of local times becomes that of a traveling 1