Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 14 (2009) > Paper 27 open journal systems 


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.


Full text: PDF

Pages: 728-751

Published on: March 11, 2009
Bibliography
  1. M. Aizenman, R. Sims, S.L. Starr. Mean-field Spin Glass Models from the Cavity-Rost Perspective. Prospects in mathematical physics 437 (2007), 1-30. Amer. Math. Soc., Providence, RI. MR2354653
  2. L.-P. Arguin, M. Aizenman. On the structure of quasi-stationary competing particles systems. arXiv: 0709.2901v1 [math.PR] (2007).
  3. E. Bolthausen, A.-S. Sznitman. On Ruelle's probability cascades and an abstract cavity method. Communications in Mathematical Physics 197 (1998), 247-276. MR1652734
  4. N.R. Chaganty, J. Sethuraman. Strong large deviation and local limit theorems. The Annals of Probability 21 (1993), 1671-1690. MR1235434
  5. D.J. Daley, D. Vere-Jones. An Introduction to the Theory of Point Processes (2003). 2nd ed. Springer, New York.
  6. A. Dembo, O. Zetouni. Large deviation techniques and applications (1998). 2nd ed. Springer, New York.
  7. J. Deny. Sur l'equation de convolution μ=μ*σ. Seminaire Brelot-Choquet-Deny. Theorie du potentiel 4 (1960), 1-11. Math. Review number not available.
  8. B. Derrida. Random-energy model: Limit of a family of disordered models. Physical Review Letters 45 (1980), 79-82. MR0575260
  9. J. Galambos. Advanced Probability Theory (1995). Marcel Dekker, Inc.
  10. M. Iltis. Sharp Asymptotics of Large Deviations in Rd, Journal of Theoretical Probability 8 (1995), 501-522. MR1340824
  11. M. Mezard, G. Parisi, M.A. Virasoro. Spin Glass Theory and Beyond (1987). World Scientific, Singapore.
  12. J. Miller. Quasi-stationary Random Overlap Structures and the Continuous Cascades. arXiv: 0806.1915v1 [math.PR] (2008).
  13. D. Ruelle. A mathematical reformulation of Derrida's REM and GREM. Communications in Mathematical Physics 108 (1987), 225-239. MR0875300
  14. A. Ruzmaikina, M. Aizenman. Characterization of invariant measures at the leading edge for competing particle systems. The Annals of Probability 338 (2005), 82-113. MR2118860
  15. M. Talagrand. Spin Glasses: A Challenge for Mathematicians (2003). Springer, Berlin.
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489