Finite-dimensional approximation of the self-diffusion coefficient for the exclusion process. (English) Zbl 1018.60097

Authors’ abstract: 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.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text: DOI


[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. · Zbl 0713.60041
[2] GIACOMIN, G., OLLA, S. and SPOHN, H. (2001). Equilibrium fluctuations for interface models. Ann. Probab. 29 1138-1172. · Zbl 1017.60100
[3] GRIGOR’YAN, A. A. (1988). On Liouville theorems for harmonic functions with finite Dirichlet integral. Math. USSR-Sb. 60 485-504. · Zbl 0646.31009
[4] KESTEN, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston. · Zbl 0522.60097
[5] KIPNIS, C. and LANDIM, C. (1999). Scaling Limit of Interacting Particle Systems. Springer, Berlin. · Zbl 0927.60002
[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. · Zbl 0588.60058
[7] LANDIM, C. (1998). Decay to equilibrium in L of asymmetric simple exclusion processes in infinite volume. Markov Process. Related Fields 4 517-534. · Zbl 0928.60093
[8] LANDIM, C. and YAU, H. T. (1997). Fluctuation-dissipation equation of asymmetric simple exclusion processes. Probab. Theory Related Fields 108 321-356. · Zbl 0884.60092
[9] LIGGETT, T. (1985). Interacting Particles Systems. Springer, Berlin. · Zbl 0559.60078
[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. · Zbl 0816.60079
[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. · Zbl 1029.60084
[12] SOARDI, P. M. (1994). Potential Theory on Infinite Networks. Lecture Notes in Math. 1590. Springer, New York. · Zbl 0818.31001
[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. · Zbl 0816.60093
[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. · Zbl 0793.60105
[15] NEW YORK, NEW YORK 10012 E-MAIL: varadhan@cims.nyu.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.