Electronic Journal of Probability

Random walk on random walks: higher dimensions

Oriane Blondel, Marcelo R. Hilário, Renato S. dos Santos, Vladas Sidoravicius, and Augusto Teixeira

Full-text: Open access

Abstract

We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 80, 33 pp.

Dates
Received: 23 October 2017
Accepted: 21 June 2019
First available in Project Euclid: 5 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1567670466

Digital Object Identifier
doi:10.1214/19-EJP337

Subjects
Primary: 60F15: Strong theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82C22: Interacting particle systems [See also 60K35] 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Keywords
random walk dynamical random environment strong law of large numbers functional central limit theorem large deviation bound renormalization regeneration times

Rights
Creative Commons Attribution 4.0 International License.

Citation

Blondel, Oriane; Hilário, Marcelo R.; dos Santos, Renato S.; Sidoravicius, Vladas; Teixeira, Augusto. Random walk on random walks: higher dimensions. Electron. J. Probab. 24 (2019), paper no. 80, 33 pp. doi:10.1214/19-EJP337. https://projecteuclid.org/euclid.ejp/1567670466


Export citation

References

  • [1] L. Avena, Random walks in dynamic random environments, Ph.D. thesis, Mathematical Institute, Faculty of Science, Leiden University, 2010.
  • [2] L. Avena, O. Blondel, and A. Faggionato, A class of random walks in reversible dynamic environments: antisymmetry and applications to the East model, J. Stat. Phys. 165 (2016), no. 1, 1–23.
  • [3] L. Avena, O. Blondel, and A. Faggionato, Analysis of random walks in dynamic random environments via l2-perturbations, Stochastic Processes and their Applications 128 (2018), no. 10, 3490–3530.
  • [4] L. Avena, T. Franco, M. Jara, and F. Völlering, Symmetric exclusion as a random environment: hydrodynamic limits, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 3, 901–916.
  • [5] L. Avena, F. den Hollander, and F. Redig, Large deviation principle for one-dimensional random walk in dynamic random environment: Attractive spin-flips and simple symmetric exclusion, Markov Processes and Related Fields 16 (2010), no. 1, 139–168.
  • [6] L. Avena, F. den Hollander, and F. Redig, Law of large numbers for a class of random walks in dynamic random environments, Electron. J. Probab. 16 (2011), no. 21, 587–617.
  • [7] L. Avena, M. Jara, and F. Völlering, Explicit ldp for a slowed rw driven by a symmetric exclusion process, Probability Theory and Related Fields 171 (2018), no. 3, 865–915.
  • [8] L. Avena, R.S. dos Santos, and F. Völlering, Transient random walk in symmetric exclusion: limit theorems and an Einstein relation, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 2, 693–709.
  • [9] J. Bérard and A. Ramírez, Fluctuations of the front in a one-dimensional model for the spread of an infection, Ann. Probab. 44 (2016), no. 4, 2770–2816.
  • [10] S.A. Bethuelsen, The contact process as seen from a random walk, ALEA, Lat. Am. J. Probab. Math. Stat. 15 (2018), 571–585.
  • [11] S.A. Bethuelsen and M. Heydenreich, Law of large numbers for random walks on attractive spin-flip dynamics, to appear in Stoch. Proc. Appl. (2014).
  • [12] S.A. Bethuelsen and F. Völlering, Absolute continuity and weak uniform mixing of random walk in dynamic random environment, Electron. J. Probab. 21 (2016), Paper No. 71, 32.
  • [13] O. Blondel, M.R. Hilario, R.S. dos Santos, V. Sidoravicius, and A. Teixeira, Random walk on random walks: low densities, ArXiv e-prints (2017).
  • [14] C. Boldrighini, I.A. Ignatyuk, V.A. Malyshev, and A. Pellegrinotti, Random walk in dynamic environment with mutual influence, Stochastic Process. Appl. 41 (1992), no. 1, 157–177.
  • [15] D. Campos, A. Drewitz, A.F. Ramírez, F. Rassoul-Agha, and T. Seppäläinen, Level 1 quenched large deviation principle for random walk in dynamic random environment, Bull. Inst. Math. Acad. Sin. (N.S.) 8 (2013), no. 1, 1–29.
  • [16] M. Hilário, F. den Hollander, R.S. dos Santos, V. Sidoravicius, and A. Teixeira, Random walk on random walks, Electron. J. Probab. 20 (2015), no. 95.
  • [17] F. den Hollander, H. Kesten, and V. Sidoravicius, Random walk in a high density dynamic random environment, Indag. Math. (N.S.) 25 (2014), no. 4, 785–799.
  • [18] F. den Hollander and R.S. dos Santos, Scaling of a random walk on a supercritical contact process, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 4, 1276–1300.
  • [19] F. den Hollander, R.S. dos Santos, and V. Sidoravicius, Law of large numbers for non-elliptic random walks in dynamic random environments, Stochastic Process. Appl. 123 (2013), no. 1, 156–190.
  • [20] M. Holmes and T.S. Salisbury, Random walks in degenerate random environments., Can. J. Math. 66 (2014), no. 5, 1050–1077.
  • [21] F. Huveneers and F. Simenhaus, Random walk driven by the simple exclusion process, Electron. J. Probab. 20 (2015), no. 105.
  • [22] H. Kesten, M.V. Kozlov, and F. Spitzer, A limit law for random walk in a random environment, Compositio Math. 30 (1975), 145–168.
  • [23] H. Kesten and V. Sidoravicius, The spread of a rumor or infection in a moving population., Ann. Probab. 33 (2005), no. 6, 2402–2462.
  • [24] H. Kesten and V. Sidoravicius, A shape theorem for the spread of an infection., Ann. Math. (2) 167 (2008), no. 3, 701–766.
  • [25] G.F. Lawler and V. Limic, Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010.
  • [26] T.M. Liggett, Interacting particle systems, Classics in Mathematics, Springer-Verlag, Berlin, 2005, Reprint of the 1985 original.
  • [27] N. Madras, A process in a randomly fluctuating environment, Ann. Probab. 14 (1986), no. 1, 119–135.
  • [28] T. Mountford and M.E. Vares, Random walks generated by equilibrium contact processes, Electron. J. Probab. 20 (2015), no. 3, 17.
  • [29] T. Orenshtein and R.S. dos Santos, Zero-one law for directional transience of one-dimensional random walks in dynamic random environments, Electron. Commun. Probab. 21 (2016), 15.
  • [30] F. Redig and F. Völlering, Random walks in dynamic random environments: a transference principle, Ann. Probab. 41 (2013), no. 5, 3157–3180.
  • [31] R.S. dos Santos, Some case studies of random walks in dynamic random environments, Ph.D. thesis, Mathematical Institute, Faculty of Science, Leiden University, 2012.
  • [32] R.S. dos Santos, Non-trivial linear bounds for a random walk driven by a simple symmetric exclusion process, Electron. J. Probab. 19 (2014), no. 49, 18.
  • [33] Ya.G. Sinaĭ, The limit behavior of a one-dimensional random walk in a random environment, Teor. Veroyatnost. i Primenen. 27 (1982), no. 2, 247–258.
  • [34] F. Solomon, Random walks in a random environment, Ann. Probab. 3 (1975), no. 1, 1–31.
  • [35] A.S. Sznitman, Topics in random walks in random environment, School and Conference on Probability Theory, ICTP Lect. Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 203–266 (electronic).
  • [36] A. Teixeira, Interlacement percolation on transient weighted graphs, Electron. J. Probab. 14 (2009), no. 54, 1604–1628.
  • [37] O. Zeitouni, Lectures on probability theory and statistics: Ecole d’eté de probabilités de saint-flour xxxi - 2001, ch. Part II: Random Walks in Random Environment, pp. 189–312, Springer Berlin Heidelberg, 2004.