• Tomasz Komorowski
  • Claudio Landim
  • Stefano Olla
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 345)


In this chapter, we prove a central limit theorem for the position of a tagged particle in exclusion processes. This problem is a special case of a random walk in a random environment. We adopt the approach of the environment as seen from the particle introduced in Sect.  1.3. It is first shown that this position can be written as the sum of a martingale and an additive functional of the exclusion process as seen from the particle. The techniques developed in the first part of the book applied to the present context permit to show that the additive functional can be itself expressed as the sum of a martingale and a remainder which vanishes asymptotically. This observation reduces the proof of the central limit theorem for the tagged particle to a central limit theorem for martingales which has been presented in the first part of the book. A variational formula for the asymptotic variance as well as bounds are given in the last section of the chapter.


Markov Process Central Limit Theorem Symmetric Part Dirichlet Form Variational Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Alexander FJ, Lebowitz JL (1990) Driven diffusive systems with a moving obstacle: a variation on the Brazil nuts problem. J Phys A, Math Gen 23:L375–L381 MathSciNetCrossRefGoogle Scholar
  2. Alexander FJ, Lebowitz JL (1994) On the drift and diffusion of a rod in a lattice fluid. J Phys A, Math Gen 27:683–696 MathSciNetzbMATHCrossRefGoogle Scholar
  3. Arratia R (1983) The motion of a tagged particle in the simple symmetric exclusion system on Z. Ann Probab 11(2):362–373 MathSciNetzbMATHCrossRefGoogle Scholar
  4. Bertini L, Toninelli C (2004) Exclusion processes with degenerate rates: convergence to equilibrium and tagged particle. J Stat Phys 117(3–4):549–580 MathSciNetzbMATHCrossRefGoogle Scholar
  5. Blumenthal RM, Getoor RK (1968) Markov processes and potential theory. Pure and applied mathematics, vol 29. Academic Press, New York zbMATHGoogle Scholar
  6. Brox T, Rost H (1984) Equilibrium fluctuations of stochastic particle systems: the role of conserved quantities. Ann Probab 12(3):742–759 MathSciNetzbMATHCrossRefGoogle Scholar
  7. Caputo P, Ioffe D (2003) Finite volume approximation of the effective diffusion matrix: the case of independent bond disorder. Ann Inst Henri Poincaré Probab Stat 39(3):505–525 MathSciNetzbMATHCrossRefGoogle Scholar
  8. Carlson JM, Grannan ER, Swindle GH (1993a) A limit theorem for tagged particles in a class of self-organizing particle systems. Stoch Process Appl 47(1):1–16 MathSciNetzbMATHCrossRefGoogle Scholar
  9. Carlson JM, Grannan ER, Swindle GH, Tour J (1993b) Singular diffusion limits of a class of reversible self-organizing particle systems. Ann Probab 21(3):1372–1393 MathSciNetzbMATHCrossRefGoogle Scholar
  10. Feller W (1971) An introduction to probability theory and its applications, vol II, 2nd edn. Wiley, New York zbMATHGoogle Scholar
  11. Gonçalves P, Jara M (2008) Scaling limits of a tagged particle in the exclusion process with variable diffusion coefficient. J Stat Phys 132(6):1135–1143 MathSciNetzbMATHCrossRefGoogle Scholar
  12. Jara M (2006) Finite-dimensional approximation for the diffusion coefficient in the simple exclusion process. Ann Probab 34(6):2365–2381 MathSciNetzbMATHCrossRefGoogle Scholar
  13. Jara M (2009) Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps. Commun Pure Appl Math 62(2):198–214 MathSciNetzbMATHCrossRefGoogle Scholar
  14. Jara MD, Landim C (2006) Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. Ann Inst Henri Poincaré Probab Stat 42(5):567–577 MathSciNetzbMATHCrossRefGoogle Scholar
  15. Jara MD, Landim C (2008) Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder. Ann Inst Henri Poincaré Probab Stat 44(2):341–361 MathSciNetzbMATHCrossRefGoogle Scholar
  16. Kipnis C (1986) Central limit theorems for infinite series of queues and applications to simple exclusion. Ann Probab 14(2):397–408 MathSciNetzbMATHCrossRefGoogle Scholar
  17. Kipnis C, Varadhan SRS (1986) Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun Math Phys 104(1):1–19 MathSciNetzbMATHCrossRefGoogle Scholar
  18. Landim C, Olla S, Volchan SB (1998a) Driven tracer particle in one-dimensional symmetric simple exclusion. Commun Math Phys 192(2):287–307 MathSciNetzbMATHCrossRefGoogle Scholar
  19. Landim C, Olla S, Varadhan SRS (2001) Symmetric simple exclusion process: regularity of the self-diffusion coefficient. Commun Math Phys 224(1):307–321. Dedicated to Joel L Lebowitz MathSciNetzbMATHCrossRefGoogle Scholar
  20. Landim C, Olla S, Varadhan SRS (2002) Finite-dimensional approximation of the self-diffusion coefficient for the exclusion process. Ann Probab 30(2):483–508 MathSciNetzbMATHCrossRefGoogle Scholar
  21. Lebowitz JL, Rost H (1994) The Einstein relation for the displacement of a test particle in a random environment. Stoch Process Appl 54(2):183–196 MathSciNetzbMATHCrossRefGoogle Scholar
  22. Liggett TM (1985) Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol 276. Springer, New York zbMATHCrossRefGoogle Scholar
  23. Loulakis M (2002) Einstein relation for a tagged particle in simple exclusion processes. Commun Math Phys 229(2):347–367 MathSciNetzbMATHCrossRefGoogle Scholar
  24. Loulakis M (2005) Mobility and Einstein relation for a tagged particle in asymmetric mean zero random walk with simple exclusion. Ann Inst Henri Poincaré Probab Stat 41(2):237–254 MathSciNetzbMATHCrossRefGoogle Scholar
  25. Osada H (1998a) An invariance principle for Markov processes and Brownian particles with singular interaction. Ann Inst Henri Poincaré Probab Stat 34(2):217–248 MathSciNetzbMATHCrossRefGoogle Scholar
  26. Osada H (1998b) Positivity of the self-diffusion matrix of interacting Brownian particles with hard core. Probab Theory Relat Fields 112(1):53–90 MathSciNetzbMATHCrossRefGoogle Scholar
  27. Owhadi H (2003) Approximation of the effective conductivity of ergodic media by periodization. Probab Theory Relat Fields 125(2):225–258 MathSciNetzbMATHCrossRefGoogle Scholar
  28. Peligrad M, Sethuraman S (2008) On fractional Brownian motion limits in one dimensional nearest-neighbor symmetric simple exclusion. ALEA Lat Am J Probab Math Stat 4:245–255 MathSciNetzbMATHGoogle Scholar
  29. Rost H, Vares ME (1985) Hydrodynamics of a one-dimensional nearest neighbor model. In: Particle systems, random media and large deviations, Brunswick, Maine, 1984. Contemp math, vol 41. Am Math Soc, Providence, pp 329–342 CrossRefGoogle Scholar
  30. Saada E (1987) A limit theorem for the position of a tagged particle in a simple exclusion process. Ann Probab 15(1):375–381 MathSciNetzbMATHCrossRefGoogle Scholar
  31. Sethuraman S (2007) On diffusivity of a tagged particle in asymmetric zero-range dynamics. Ann Inst Henri Poincaré Probab Stat 43(2):215–232 MathSciNetzbMATHCrossRefGoogle Scholar
  32. Sethuraman S, Varadhan SRS, Yau HT (2000) Diffusive limit of a tagged particle in asymmetric simple exclusion processes. Commun Pure Appl Math 53(8):972–1006 MathSciNetzbMATHCrossRefGoogle Scholar
  33. Shiga T (1988) Tagged particle motion in a clustered random walk system. Stoch Process Appl 30(2):225–252 MathSciNetzbMATHCrossRefGoogle Scholar
  34. Spitzer F (1970) Interaction of Markov processes. Adv Math 5:246–290 MathSciNetzbMATHCrossRefGoogle Scholar
  35. Szász D, Tóth B (1987a) A dynamical theory of Brownian motion for the Rayleigh gas. In: Proceedings of the symposium on statistical mechanics of phase transitions—mathematical and physical aspects, Trebon, 1986, vol 47, pp 681–693 Google Scholar
  36. Szász D, Tóth B (1987b) Towards a unified dynamical theory of the Brownian particle in an ideal gas. Commun Math Phys 111(1):41–62 zbMATHCrossRefGoogle Scholar
  37. Tanemura H (1989) Ergodicity for an infinite particle system in R d of jump type with hard core interaction. J Math Soc Jpn 41(4):681–697 MathSciNetzbMATHCrossRefGoogle Scholar
  38. Toninelli C, Biroli G (2004) Dynamical arrest, tracer diffusion and kinetically constrained lattice gases. J Stat Phys 117(1–2):27–54 MathSciNetzbMATHCrossRefGoogle Scholar
  39. Varadhan SRS (1995) Self-diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion. Ann Inst Henri Poincaré Probab Stat 31(1):273–285 MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tomasz Komorowski
    • 1
    • 2
  • Claudio Landim
    • 3
    • 4
  • Stefano Olla
    • 5
  1. 1.Institute of MathematicsMaria Curie-Skłodowska UniversityLublinPoland
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  3. 3.Instituto de Matemática (IMPA)Rio de JaneiroBrazil
  4. 4.CNRS UMR 6085Université de RouenSaint-Étienne-du-RouvrayFrance
  5. 5.CEREMADE, CNRS UMR 7534Université Paris-DauphineParisFrance

Personalised recommendations