Scaling of a random walk on a supercritical contact process

F. den Hollander; R. S. dos Santos

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1276-1300
  • ISSN: 0246-0203

Abstract

top
We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle under the annealed law. The qualitative dependence of the asymptotic speed and the volatility on the infection parameter is investigated, and some open problems are mentioned.

How to cite

top

den Hollander, F., and dos Santos, R. S.. "Scaling of a random walk on a supercritical contact process." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1276-1300. <http://eudml.org/doc/272085>.

@article{denHollander2014,
abstract = {We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle under the annealed law. The qualitative dependence of the asymptotic speed and the volatility on the infection parameter is investigated, and some open problems are mentioned.},
author = {den Hollander, F., dos Santos, R. S.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk; dynamic random environment; contact process; strong law of large numbers; functional central limit theorem; large deviation principle; Space-time cones; clusters of infections; coupling; regeneration times; supercritical contact process; space-time cones},
language = {eng},
number = {4},
pages = {1276-1300},
publisher = {Gauthier-Villars},
title = {Scaling of a random walk on a supercritical contact process},
url = {http://eudml.org/doc/272085},
volume = {50},
year = {2014},
}

TY - JOUR
AU - den Hollander, F.
AU - dos Santos, R. S.
TI - Scaling of a random walk on a supercritical contact process
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1276
EP - 1300
AB - We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle under the annealed law. The qualitative dependence of the asymptotic speed and the volatility on the infection parameter is investigated, and some open problems are mentioned.
LA - eng
KW - random walk; dynamic random environment; contact process; strong law of large numbers; functional central limit theorem; large deviation principle; Space-time cones; clusters of infections; coupling; regeneration times; supercritical contact process; space-time cones
UR - http://eudml.org/doc/272085
ER -

References

top
  1. [1] L. Avena. Random walks in dynamic random environments. Ph.D. thesis, Leiden Univ., 2010. Zbl1246.82040
  2. [2] L. Avena. Symmetric exclusion as a model of non-elliptic dynamical random conductances. Electron. Commun. Probab. 17 (2012). Zbl1252.60097MR2988390
  3. [3] 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 Process. Related Fields16 (2010) 139–168. Zbl1198.82049MR2664339
  4. [4] 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) 587–617. Zbl1228.60113MR2786643
  5. [5] L. Avena, R. dos Santos and F. Völlering. Transient random walk in symmetric exclusion: Limit theorems and an Einstein relation. Available at arXiv:1102.1075 [math.PR]. Zbl1277.60185
  6. [6] C. Bezuidenhout and G. Grimmett. Exponential decay for subcritical contact and percolation processes. Ann. Probab.19 (1991) 984–1009. Zbl0743.60107MR1112404
  7. [7] M. Birkner, J. Černý, A. Depperschmidt and N. Gantert. Directed random walk on an oriented percolation cluster. Available at arXiv:1204.2951 [math.PR]. Zbl1326.60142
  8. [8] F. den Hollander, H. Kesten and V. Sidoravicius. Directed random walk on the backbone of an oriented percolation cluster. Available at arXiv:1305.0923 [math.PR]. 
  9. [9] R. Durrett and R. H. Schonmann. Large deviations for the contact process and two-dimensional percolation. Probab. Theory Related Fields77 (1988) 583–603. Zbl0621.60108MR933991
  10. [10] P. Glynn and W. Whitt. Large deviations behavior of counting processes and their inverses. Queueing Syst.17 (1994) 107–128. Zbl0805.60023MR1295409
  11. [11] T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York, 1985. Zbl0559.60078MR776231
  12. [12] T. M. Liggett. An improved subadditive ergodic theorem. Ann. Probab.13 (1985) 1279–1285. Zbl0579.60023MR806224
  13. [13] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften 324. Springer, Berlin, 1999. Zbl0949.60006MR1717346
  14. [14] T. M. Liggett and J. Steif. Stochastic domination: The contact process, Ising models and FKG measures. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 223–243. Zbl1087.60074MR2199800
  15. [15] A.-S. Sznitman. Lectures on random motions in random media. In Ten Lectures on Random Media 1851–1869. DMV-Lectures 32. Birkhäuser, Basel, 2002. MR1890289
  16. [16] J. van den Berg, O. Häggström and J. Kahn. Some conditional correlation inequalities for percolation and related processes. Random Structures Algorithms29 (2006) 417–435. Zbl1112.60087MR2268229
  17. [17] O. Zeitouni. Random walks in random environment. In XXXI Summer School in Probability, Saint-Flour, 2001193–312. Lecture Notes in Math. 1837. Springer, Berlin, 2004. Zbl1060.60103MR2071631

NotesEmbed ?

top

You must be logged in to post comments.