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 Electronic Communications in Probability > Vol. 15(2010) > Paper 22 open journal systems 


A limit theorem for particle current in the symmetric exclusion process

Alexander Vandenberg-Rodes, UC Los Angeles


Abstract
Using the recently discovered strong negative dependence properties of the symmetric exclusion process, we derive general conditions for when the normalized current of particles between regions converges to the Gaussian distribution. The main novelty is that the results do not assume any translation invariance, and hold for most initial configurations.


Full text: PDF

Pages: 240-252

Published on: June 28, 2010


Bibliography
  1. Andjel, Enrique D. A correlation inequality for the symmetric exclusion process. Ann. Probab. 16 (1988), no. 2, 717--721. MR0929073 (89d:60192)
  2. Arratia, Richard. The motion of a tagged particle in the simple symmetric exclusion system on $Z$. Ann. Probab. 11 (1983), no. 2, 362--373. MR0690134 (84g:60156)
  3. Balázs, M.; Seppäläinen, T. Order of current variance and diffusivity in the asymmetric simple exclusion process (2008), arXiv:math/0608400
  4. Borcea, Julius; Brändén, Petter; Liggett, Thomas M. Negative dependence and the geometry of polynomials. J. Amer. Math. Soc. 22 (2009), no. 2, 521--567. MR2476782 (2010b:62215)
  5. De Masi, A.; Ferrari, P. A. Flux fluctuations in the one dimensional nearest neighbors symmetric simple exclusion process. J. Statist. Phys. 107 (2002), no. 3-4, 677--683. MR1898853 (2003b:82041)
  6. Derrida, Bernard; Gerschenfeld, Antoine. Current fluctuations of the one dimensional symmetric simple exclusion process with step initial condition. J. Stat. Phys. 136 (2009), no. 1, 1--15. MR2525223
  7. Erdős, P.; Rényi, A. On Cantor's series with convergent $sum 1/qsb{n}$. Ann. Univ. Sci. Budapest. Eötvös. Sect. Math. 2 1959 93--109. MR0126414 (23 #A3710)
  8. Ferrari, Patrik L.; Spohn, Herbert. Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265 (2006), no. 1, 1--44. MR2217295 (2007g:82038a)
  9. Hough, J. Ben; Krishnapur, Manjunath; Peres, Yuval; Virág, Bálint. Determinantal processes and independence. Probab. Surv. 3 (2006), 206--229 (electronic). MR2216966 (2006m:60068)
  10. Kawazu, Kiyoshi; Kesten, Harry. On birth and death processes in symmetric random environment. J. Statist. Phys. 37 (1984), no. 5-6, 561--576. MR0775792 (86g:60101)
  11. Jara, M. D.; Landim, C. Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 5, 567--577. MR2259975 (2008h:60406)
  12. Jara, M. D.; Landim, C. Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 2, 341--361. MR2446327 (2010f:60278)
  13. Levin, B. Ja. Distribution of zeros of entire functions. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. Revised edition. Translations of Mathematical Monographs, 5. American Mathematical Society, Providence, R.I., 1980. xii+523 pp. ISBN: 0-8218-4505-5 MR0589888 (81k:30011)
  14. Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4 MR0776231 (86e:60089)
  15. Liggett, T. M. Negative correlations and particle systems. Markov Process. Related Fields 8 (2002), no. 4, 547--564. MR1957219 (2004f:60207)
  16. Liggett, T. M. Distributional limits for the symmetric exclusion process Stoch. Proc. App. 119 (2009)1--15. Math. Review number not available.
  17. Lyons, Russell. Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. No. 98 (2003), 167--212. MR2031202 (2005b:60024)
  18. Peligrad, Magda; Sethuraman, Sunder. On fractional Brownian motion limits in one dimensional nearest-neighbor symmetric simple exclusion. ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 245--255. MR2448774 (2009j:60209)
  19. Pemantle, Robin. Towards a theory of negative dependence. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41 (2000), no. 3, 1371--1390. MR1757964 (2001g:62039)
  20. Petrov, V. V. Sums of independent random variables. Translated from the Russian by A. A. Brown. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer-Verlag, New York-Heidelberg, 1975. x+346 pp. MR0388499 (52 #9335)
  21. Tracy, Craig A.; Widom, Harold. Asymptotics in ASEP with step initial condition. Comm. Math. Phys. 290 (2009), no. 1, 129--154. MR2520510 (2010f:60283)
















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Electronic Communications in Probability. ISSN: 1083-589X