Journal of Statistical Physics

, Volume 127, Issue 2, pp 431–455 | Cite as

Exact Connections between Current Fluctuations and the Second Class Particle in a Class of Deposition Models

  • Márton Balázs
  • Timo Seppäläinen


We consider a large class of nearest neighbor attractive stochastic interacting systems that includes the asymmetric simple exclusion, zero range, bricklayers’ and the symmetric K-exclusion processes. We provide exact formulas that connect particle flux (or surface growth) fluctuations to the two-point function of the process and to the motion of the second class particle. Such connections have only been available for simple exclusion where they were of great use in particle current fluctuation investigations.


simple exclusion zero range bricklayers’, current fluctuations second class particle space-time covariance two-point function diffusivity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. D. Andjel, Invariant measures for the zero range process. Ann. Probab. 10(3):325–547 (1982).MathSciNetGoogle Scholar
  2. 2.
    M. Balázs, Growth fluctuations in a class of deposition models. Ann. Inst. H. Poincaré Probab. Statist. 39:639–685 (2003).zbMATHCrossRefGoogle Scholar
  3. 3.
    M. Balázs, E. Cator and T. Seppäläinen, Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11:1094–1132 (2006).MathSciNetGoogle Scholar
  4. 4.
    M. Balázs, F. Rassoul-Agha, T. Seppäläinen, and S. Sethuraman, Existence of the zero range process and a deposition model with superlinear growth rates. To appear in Ann. Probab., http:// (2006).
  5. 5.
    M. Balázs and T. Seppäläinen, Order of current variance and diffusivity in the asymmetric simple exclusion process. Submitted, (2006).
  6. 6.
    L. Booth, Random Spatial Structures and Sums. PhD thesis, Utrecht University (2002).Google Scholar
  7. 7.
    C. Cocozza-Thivent, Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70:509–523 (1985).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    P. A. Ferrari and L. R. G. Fontes, Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22:820–832 (1994).zbMATHMathSciNetGoogle Scholar
  9. 9.
    P. L. Ferrari and H. Spohn, Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265(1):1–44 (2006).zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    T. M. Liggett, An infinite particle system with zero range interactions. Ann. Probab. 1(2):240–253 (1973).zbMATHMathSciNetGoogle Scholar
  11. 11.
    T. M. Liggett, Interacting Particle Systems (Springer-Verlag, 1985).Google Scholar
  12. 12.
    M. Prähofer and H. Spohn, Current fluctuations for the totally asymmetric simple exclusion process. In In and out of equilibrium (Mambucaba, 2000), Vol. 51 of Progr. Probab. (Birkhäuser Boston, Boston, MA, 2002, pp. 185–204).Google Scholar
  13. 13.
    C. Quant, On the construction and stationary distributions of some spatial queueing and particle systems. PhD thesis, Utrecht University, 2002.Google Scholar
  14. 14.
    J. Quastel and B. Valkó, t1/3 Superdiffusivity of finite-range asymmetric exclusion processes on ℤ. To appear in Comm. Math. Phys. (2006).
  15. 15.
    F. Rezakhanlou, Hydrodynamic limit for attractive particle systems on ℤd. Comm. Math. Phys. 140(3):417–448 (1991).zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    F. Spitzer, Interaction of Markov processes. Adv. Math. 5:246–290 (1970).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    H. Spohn, Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics (Springer Verlag, Heidelberg, 1991).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Sztochasztika TanszékMatematika IntézetBudapestHungary

Personalised recommendations