Competing Particle Systems Evolving by I.I.D. Increments
Mykhaylo Shkolnikov, Stanford University
Abstract
We consider competing particle systems in Rd, i.e. random locally finite upper bounded
configurations of points in Rd evolving in discrete time steps. In each step i.i.d. increments are added to
the particles independently of the initial configuration and the previous steps. Ruzmaikina and Aizenman characterized
quasi-stationary measures of such an evolution, i.e. point processes for which the joint distribution of the gaps between the
particles is invariant under the evolution, in case d=1 and restricting to increments having a density and an everywhere
finite moment generating function. We prove corresponding versions of their theorem in dimension d=1 for heavy-tailed
increments in the domain of attraction of a stable law and in dimension d>1 for lattice type increments with an
everywhere finite moment generating function. In all cases we only assume that under the initial configuration no two
particles are located at the same point. In addition, we analyze the attractivity of quasi-stationary Poisson point processes
in the space of all Poisson point processes with almost surely infinite, locally finite and upper bounded configurations.
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