The Annals of Probability

Fluctuations in the occupation time of a site in the asymmetric simple exclusion process

Cédric Bernardin

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Abstract

We consider the simple asymmetric exclusion process with nonzero drift under the stationary Bernoulli product measure at density $\rho$. We prove that for dimension $d=2$ the occupation time of the site 0 is diffusive as soon as $\rho\neq 1/2$. For dimension $d=1$, if the density $\rho$ is equal to $1/2$, we prove that the time t variance of the occupation time of the site 0 diverges in a certain sense at least as $t^{5/4}$.

Article information

Source
Ann. Probab., Volume 32, Number 1B (2004), 855-879.

Dates
First available in Project Euclid: 11 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1079021466

Digital Object Identifier
doi:10.1214/aop/1079021466

Mathematical Reviews number (MathSciNet)
MR2039945

Zentralblatt MATH identifier
1071.60096

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F05: Central limit and other weak theorems

Keywords
Exclusion process occupation time of a site invariance principle

Citation

Bernardin, Cédric. Fluctuations in the occupation time of a site in the asymmetric simple exclusion process. Ann. Probab. 32 (2004), no. 1B, 855--879. doi:10.1214/aop/1079021466. https://projecteuclid.org/euclid.aop/1079021466


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References

  • Baik, J. and Rains, E. M. (2000). Limiting distributions for a polynuclear growth model with external sources. J. Statist. Phys. 100 523--542.
  • Kipnis, C. (1987). Fluctuations des temps d'occupation d'un site dans l'exclusion simple symétrique. Ann. Inst. H. Poincaré Probab. Statist. 23 21--35.
  • Kipnis, C. and Varadhan, S. R. S. (1986). Central limit for additive functionals of reversible Markov processes. Comm. Math. Phys. 104 1--19.
  • Landim, C., Quastel, J., Salmhofer, M. and Yau, H. T. (2002). Superdiffusivity of asymmetric exclusion process in dimension $1$ and $2$. Preprint arXiv:math.PR/0201317.
  • Landim, C. and Yau, H. T. (1997). Fluctuation--dissipation of asymmetric simple exclusion processes. Probab. Theory Related Fields 108 321--356.
  • Prähofer, M. and Spohn, H. (2002). Current fluctuations for the totally asymmetric simple exclusion process. In In and Out of Equilibrium: Probability with a Physics Flavor (V. Sidoravicius, ed.) 185--204. Birkhäuser, Boston.
  • Salmhofer, M. and Yau, H. T. (2001). Superdiffusivity of the two dimensional asymmetric simple exclusion process. Preprint.
  • Seppäläinen, T. and Sethuraman, S. (2003). Transience of second-class particles and diffusive bounds for additive functionals in one-dimensional asymmetric exclusion processes. Ann. Probab. 31 148--169.
  • Sethuraman, S. (2000). Central limit theorems for additive functionals of the simple exclusion process. Ann. Probab. 28 277--302.
  • Sethuraman, S. (2003). An equivalence of $H_-1$ norms for the simple exclusion process. Ann. Probab. 31 35--62.
  • Sethuraman, S., Varadhan, S. R. S. and Yau, H. T. (2000). Diffusive limit of a tagged particle in asymmetric simple exclusion process. Comm. Pure Appl. Math. 53 972--1006.
  • Varadhan, S. R. S. (1995). Self diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion. Ann. Inst. H. Poincaré Probab. Statist. 31 273--285.
  • Yau, H. T. (2002). $(\log t)^2/3$ law of the two dimensional asymmetric simple exclusion process. Preprint arXiv:math-ph/0201057 v1.