Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 15(2010) > Paper 10 open journal systems 


Hydrodynamic limit of zero range processes among random conductances on the supercritical percolation cluster

Alessandra Faggionato, Department of Mathematics. University La Sapienza, Rome. Italy


Abstract
We consider i.i.d. random variables omega={omega(b)} parameterized by the family of bonds in Z^d, d>1. The random variable omega(b) is thought of as the conductance of bond b and it ranges in a finite interval [0,c_0]. Assuming the probability of the event {omega(b)>0} to be supercritical and denoting by C(omega) the unique infinite cluster associated to the bonds with positive conductance, we study the zero range process on C(omega) with omega(b)-proportional probability rate of jumps along bond b. For almost all realizations of the environment we prove that the hydrodynamic behavior of the zero range process is governed by a nonlinear heat equation, independent from omega. As byproduct of the above result and the blocking effect of the finite clusters, we discuss the bulk behavior of the zero range process on Z^d with conductance field omega. We do not require any ellipticity condition.


Full text: PDF

Pages: 259-291

Published on: March 30, 2010
Bibliography
  1. E. D. Andjel. Invariant measures for the zero range process. Ann. Probab. 10 (1982), 525-547. Math. Review 83j:60106
  2. P. Antal, A. Pisztora. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996), 1036-1048. Math. Review 98b:60168
  3. D. Ben-Avraham, S. Havlin. Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge (2000). Math. Review 2003h:82001
  4. O. Benois, C.Kipnis, C. Landim. Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes. Stochastic Process. Appl. 55 (1995), 65-89. Math. Review 96a:60077.
  5. H. Brézis, M.G. Crandall. Uniqueness of solutions of the initial--value problem for u_t- ΔΦ(u)=0. J. Math. Pures and appl. 58 (1979), 153-163. Math. Review 80e:35029
  6. N. Berger, M. Biskup. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (2007), 83-120. Math. Review 2007m:60085
  7. M. Biskup, T.M. Prescott. Functional CLT for random walk among bounded random conductances. Electronic Journal of Probability 12 (2007), 1323-1348. Math. Review 2009d:60336
  8. C-C. Chang, C. Landim, S. Olla. Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d≥3. Probab. Theory Relat. Fields 119 (2001), 381–409. Math. Review 2002e:60157
  9. A. De Masi, P. Ferrari, S. Goldstein, W.D. Wick. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statis. Phys. 55 (1985), 787-855. Math. Review 91e:60107
  10. J.D. Deuschel, A. Pisztora. Surface order large deviations for high--density percolation. Probab. Theory Related Fields 104 (1996), 467-482. Math. Review 97d:60053
  11. A. Faggionato. Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit. Electron. J. Probab. 13 (2008), 2217-2247. Math. Review number not available
  12. A. Faggionato. Bulk diffusion of 1D exclusion process with bond disorder. Markov Processes and Related Fields 13 (2007), 519-542. Math. Review 2008j:60230
  13. A. Faggionato, M. Jara, C. Landim. Hydrodynamic limit of one dimensional subdiffusive exclusion processes with random conductances. Probab. Theory Related Fields 144 (2009), 633-667. Math. Review number not available
  14. A. Faggionato, F. Martinelli. Hydrodynamic limit of a disordered lattice gas. Probab. Theory and Related Fields 127 (2003), 535-608. Math. Review 2004i:82041
  15. J. Fritz. Hydrodynamics in a symmetric random medium. Comm. Math. Phys. 125 (1989), 13-25. Math. Review 91c:82060
  16. L. Fontes, C.M. Newman. First Passage Percolation for Random Colorings of Z^d. Ann. Appl. Probab. 3 (1993), 746-762. Math. Review 94k:60156a
  17. L. Fontes, C.M. Newman. Correction: First Passage Percolation for Random Colorings of Z^d. Ann. Appl. Probab. 4 (1994), 254-254. Math. Review 94k:60156b
  18. P. Goncalves, M. Jara. Scaling limit of gradient systems in random environment. J. Stat. Phys. 131 (2008), 691-716. Math. Review 2009d:82068
  19. P.Goncalves, M. Jara. Density fluctuations for a zero-range process on the percolation cluster. Electron. Commun. Probab. 14 (2009), 382-395. Math. Review number not available
  20. G. Grimmett. Percolation. Second edition. Springer, Berlin (1999). Math. Review 2001a:60114
  21. M. Jara. Hydrodynamic limit for a zero-range process in the Sierpinski gasket. Comm. Math. Phys. 288 (2009) 773-797. Math. Review number not available
  22. M. Jara, C. Landim. Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. Ann. Inst. H. Poincaré, Prob. et Stat. 42 (2006), 567-577. Math. Review 2008h:60406
  23. H. Kesten. Percolation theory for mathematicians. Progress in Probability and Statistics, Vol. 2, Birkhauser, Boston, (1982). Math. Review 84i:60145
  24. C. Kipnis, C. Landim. Scaling limits of interacting particle systems. Springer, Berlin (1999). Math. Review 2000i:60001
  25. C. Kipnis, S.R.S. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Commun. Math. Phys. 104 (1986), 1-19. Math. Review 87i:60038
  26. C. Landim, T. Franco. Hydrodynamic limit of gradient exclusion processes with conductances. Archive for Rational Mechanics and Analysis 195 (2010), 409-439. Math. Review number not available
  27. C. Landim, M. Mourragui. Hydrodynamic limit of mean zero asymmetric zero range processes in infinite volume. Ann. Inst. H. Poincaré, Prob. et Stat. 33 (1997), 65-82. Math. review 98e:60159
  28. T.M. Liggett. Interacting particle systems. Springer, New York (1985). Math. Review 86e:60089
  29. P. Mathieu. Quenched invariance principles for random walks with random conductances}. J. Stat. Phys. 130 (2008), 1025-1046. Math. Review 2009b:82040
  30. P. Mathieu, A.L. Piatnitski. Quenched invariance principles for random walks on percolation clusters}. Proceedings of the Royal Society A 463 (2007), 2287-2307. Math. Review 2008e:82033
  31. J. Quastel. Diffusion in disordered media. In Proceedings in Nonlinear Stochastic PDEs (1996), T. Funaki and W. Woyczinky eds, Springer, New York, 65-79. Math. Review 97k:60278
  32. J. Quastel. Bulk diffusion in a system with site disorder. Ann. Probab. 34 (2006), 1990-2036. Math. Review 2007i:60133
  33. T. Seppäläinen. Translation Invariant Exclusion Processes. Available online
  34. V. Sidoravicius, A.-S. Sznitman. Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 (2004), 219–244. Math. Review 2005d:60155
  35. F.J. Valentim. Hydrodynamic limit of gradient exclusion processes with conductances on Z^d. Preprint (2009).
  36. S.R.S. Varadhan. Nonlinear diffusion limit for a system with nearest neighbor interactions II. Asymptotic Problems in Probability Theory: Stochastic Models and Diffusion on Fractals , edited by K. Elworthy and N. Ikeda, Pitman Research Notes in Mathematics 283 , Wiley, 75-128 (1994). Math. Review 97a:60144
  37. J.L. Vázquez. The porous medium equation: mathematical theory. Clarendon Press, Oxford (2007). Math. Review 2008e:35003
















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