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2009 Competing Particle Systems Evolving by I.I.D. Increments
Mykhaylo Shkolnikov
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Electron. J. Probab. 14: 728-751 (2009). DOI: 10.1214/EJP.v14-635


We consider competing particle systems in $\mathbb{R}^d$, i.e. random locally finite upper bounded configurations of points in $\mathbb{R}^d$ 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 $d\geq 1$ 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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Mykhaylo Shkolnikov. "Competing Particle Systems Evolving by I.I.D. Increments." Electron. J. Probab. 14 728 - 751, 2009.


Accepted: 11 March 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1190.60039
MathSciNet: MR2486819
Digital Object Identifier: 10.1214/EJP.v14-635

Primary: 60G55
Secondary: 60G70 , 60K40 , 62P35

Keywords: Competing particle systems , large deviations , Spin glasses

Vol.14 • 2009
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