The Annals of Probability

Finite-dimensional approximation of the self-diffusion coefficient for the exclusion process

C. Landim, S. Olla, and R. S. Varadhan

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We show that for the symmetric simple exclusion process on $/mathbb{Z}^d$ the self-diffusion coefficient of a tagged particle is stable when approximated by simple exclusion processes on large periodic lattices. The proof depends on a similar stability property of the asymptotic variance of additive functionals of mean 0. This requires establishing a property for the Dirichlet space known as the Liouville-D property.

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Ann. Probab. Volume 30, Number 2 (2002), 483-508.

First available in Project Euclid: 7 June 2002

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

self-diffusion tagged particle exclusion process Liouville property


Landim, C.; Olla, S.; Varadhan, R. S. Finite-dimensional approximation of the self-diffusion coefficient for the exclusion process. Ann. Probab. 30 (2002), no. 2, 483--508. doi:10.1214/aop/1023481000.

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  • [1] DE MASI, A., FERRARI, P. A., GOLDSTEIN, S. and WICK, W. D. (1989). An invariance principle for reversible Markov processes: applications to random motions in random environments. J. Statist. Phys. 55 787-855.
  • [2] GIACOMIN, G., OLLA, S. and SPOHN, H. (2001). Equilibrium fluctuations for interface models. Ann. Probab. 29 1138-1172.
  • [3] GRIGOR'YAN, A. A. (1988). On Liouville theorems for harmonic functions with finite Dirichlet integral. Math. USSR-Sb. 60 485-504.
  • [4] KESTEN, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston.
  • [5] KIPNIS, C. and LANDIM, C. (1999). Scaling Limit of Interacting Particle Systems. Springer, Berlin.
  • [6] KIPNIS, C. and VARADHAN, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Comm. Math. Phys. 106 1-19.
  • [7] LANDIM, C. (1998). Decay to equilibrium in L of asymmetric simple exclusion processes in infinite volume. Markov Process. Related Fields 4 517-534.
  • [8] LANDIM, C. and YAU, H. T. (1997). Fluctuation-dissipation equation of asymmetric simple exclusion processes. Probab. Theory Related Fields 108 321-356.
  • [9] LIGGETT, T. (1985). Interacting Particles Systems. Springer, Berlin.
  • [10] OSADA, H. and SAITOH, T. (1995). An invariance principle for non-symmetric Markov processes and reflecting diffusions in random domains. Probab. Theory Related Fields 101 45-63.
  • [11] SETHURAMAN, S., VARADHAN, S. R. S. and YAU, H. T. (2000). Diffusive limit of a tagged particle in asymmetric exclusion process. Comm. Pure Appl. Math. 53 972-1006.
  • [12] SOARDI, P. M. (1994). Potential Theory on Infinite Networks. Lecture Notes in Math. 1590. Springer, New York.
  • [13] VARADHAN, S. R. S. (1995). Self diffusion of a tagged particle in equilibrium for asymmetric mean zero random walks with simple exclusion. Ann. Inst. H. Poincaré Probab. Statist. 31 273-285.
  • [14] VARADHAN, S. R. S. (1994). Non-linear diffusion limit for a system with nearest neighbor interactions II. In Asymptotic Problems in Probability Theory: Stochastic Models and Diffusion on Fractals (K. D. Elworthy and N. Ikeda, eds.). 75-128. Wiley, New York.