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Journal of Statistical Physics

, Volume 171, Issue 4, pp 656–678 | Cite as

Random Walk on a Perturbation of the Infinitely-Fast Mixing Interchange Process

  • Michele Salvi
  • François Simenhaus
Article

Abstract

We consider a random walk in dimension \(d\ge 1\) in a dynamic random environment evolving as an interchange process with rate \(\gamma >0\). We prove that, if we choose \(\gamma \) large enough, almost surely the empirical velocity of the walker \(X_t/t\) eventually lies in an arbitrary small ball around the annealed drift. This statement is thus a perturbation of the case \(\gamma =+\infty \) where the environment is refreshed between each step of the walker. We extend three-way part of the results of Huveneers and Simenhaus (Electron J Probab 20(105):42, 2015), where the environment was given by the 1-dimensional exclusion process: (i) We deal with any dimension \(d\ge 1\); (ii) We treat the much more general interchange process, where each particle carries a transition vector chosen according to an arbitrary law \(\mu \); (iii) We show that \(X_t/t\) is not only in the same direction of the annealed drift, but that it is also close to it.

Keywords

Random walk Dynamic random environment Interchange process Limit theorems Renormalisation 

Mathematics Subject Classification

60K37 82C22 60Fxx 82D30 

Notes

Acknowledgements

We would like to thank an anonymous referee for pointing out useful remarks. M. Salvi was financially supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant agreement No. 656047. F. Simenhaus research was supported by the ANR-15-CE40-0020-01 Grant LSD and the ANR/FNS-16-CE93-0003 Grant MALIN.

References

  1. 1.
    Avena, L., Blondel, O., Faggionato, A.: Analysis of random walks in dynamic random environments via \(l^2\)-perturbations. arXiv:1602.06322 (2016)
  2. 2.
    Avena, L., Blondel, O., Faggionato, A.: A class of random walks in reversible dynamic environments: antisymmetry and applications to the East model. J. Stat. Phys. 165(1), 1–23 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avena, L., den Hollander, F., Redig, F.: Law of large numbers for a class of random walks in dynamic random environments. Electron. J. Probab. 16(21), 587–617 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Avena, L., dos Santos, R.S., Völlering, F.: Transient random walk in symmetric exclusion: limit theorems and an Einstein relation. ALEA Lat. Am. J. Probab. Math. Stat. 10(2), 693–709 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Berger, N., Cohen, M., Rosenthal, R.: Local limit theorem and equivalence of dynamic and static points of view for certain ballistic random walks in i.i.d. environments. Ann. Probab. 44(4), 2889–2979 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blondel, O., Hilário, M.R., Dos Santos, R.S., Sidoravicius, V., Teixeira, A.: Random walk on random walks: higher dimensions. arXiv:1709.01253 (2017)
  7. 7.
    Blondel, O., Hilário, M.R., Dos Santos, R.S., Sidoravicius, V., Teixeira, A.: Random walk on random walks: low densities. arXiv:1709.01257 (2017)
  8. 8.
    Berger, N., Salvi, M.: On the speed of random walks among random conductances. ALEA Lat. Am. J. Probab. Math. Stat. 10(2), 1063–1083 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bandyopadhyay, A., Zeitouni, O.: Random walk in dynamic Markovian random environment. ALEA Lat. Am. J. Probab. Math. Stat. 1, 205–224 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    den Hollander, F., dos Santos, R.S.: Scaling of a random walk on a supercritical contact process. Ann. Inst. Henri Poincaré Probab. Stat. 50(4), 1276–1300 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    den Hollander, F., dos Santos, R.S., Sidoravicius, V.: Law of large numbers for non-elliptic random walks in dynamic random environments. Stoch. Process. Appl. 123(1), 156–190 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dolgopyat, D., Keller, G., Liverani, C.: Random walk in Markovian environment. Ann. Probab. 36(5), 1676–1710 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Goldschmidt, C., Ueltschi, D., Windridge, P.: Quantum Heisenberg models and their probabilistic representations. In Entropy and the Quantum II. Contemporary Mathematics, vol. 552, pp. 177–224. American Mathematical Society, Providence, RI (2011)Google Scholar
  14. 14.
    Hilário, M.R., den Hollander, F., dos Santos, R.S., Sidoravicius, V., Teixeira, A.: Random walk on random walks. Electron. J. Probab 20(95), 35 (2015)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Huveneers, F., Simenhaus, F.: Random walk driven by the simple exclusion process. Electron. J. Probab 20(105), 42 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Liggett, T.M.: Interacting Particle Systems. Classics in Mathematics. Springer-Verlag, Berlin (2005). Reprint of the 1985 originalCrossRefzbMATHGoogle Scholar
  17. 17.
    Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics, vol. 123. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  18. 18.
    Redig, F., Völlering, F.: Random walks in dynamic random environments: a transference principle. Ann. Probab. 41(5), 3157–3180 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Spohn, H.: Tracer diffusion in lattice gases. J. Stat. Phys. 59(5), 1227–1239 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tóth, B.: Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet. Lett. Math. Phys. 28(1), 75–84 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université Paris-Dauphine, PSL Research University, CNRS, UMR [7534], CEREMADEParisFrance
  2. 2.Université Paris-Dauphine, CNRS, UMR [7534], CEREMADEParisFrance

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