Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling of a random walk on a supercritical contact process

F. den Hollander and R. S. dos Santos

Full-text: Open access

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.

Résumé

Nous prouvons une loi forte des grands nombres pour une marche aléatoire dans un milieu aléatoire dynamique donné par un processus de contact sur-critique unidimensionnel en équilibre. La preuve utilise un argument de couplage basé sur l’observation que la marche est finalement confinée dans l’union de cônes spatio-temporels inclus dans les clusters d’infection générés par des infections individuelles. Si les taux locaux de saut de la marche sont plus petits que la vitesse de propagation de l’infection, la marche est finalement confinée dans un seul cône, ce qui entraîne l’existence de temps de régénération en lesquels la marche oublie son passé. Ces temps de régénération sont utilisés pour prouver un théorème central limite fonctionnel et un principe de grandes déviations sous la loi “annealed.”

La dépendance de la vitesse et de la variance asymptotiques par rapport au paramètre d’infection est étudiée, et quelques problèmes ouverts sont mentionnés.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 4 (2014), 1276-1300.

Dates
First available in Project Euclid: 17 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1413555500

Digital Object Identifier
doi:10.1214/13-AIHP561

Mathematical Reviews number (MathSciNet)
MR3269994

Zentralblatt MATH identifier
1303.31008

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 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

Citation

den Hollander, F.; dos Santos, R. S. Scaling of a random walk on a supercritical contact process. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 4, 1276--1300. doi:10.1214/13-AIHP561. https://projecteuclid.org/euclid.aihp/1413555500


Export citation

References

  • [1] L. Avena. Random walks in dynamic random environments. Ph.D. thesis, Leiden Univ., 2010.
  • [2] L. Avena. Symmetric exclusion as a model of non-elliptic dynamical random conductances. Electron. Commun. Probab. 17 (2012).
  • [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 Fields 16 (2010) 139–168.
  • [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.
  • [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].
  • [6] C. Bezuidenhout and G. Grimmett. Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19 (1991) 984–1009.
  • [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].
  • [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] R. Durrett and R. H. Schonmann. Large deviations for the contact process and two-dimensional percolation. Probab. Theory Related Fields 77 (1988) 583–603.
  • [10] P. Glynn and W. Whitt. Large deviations behavior of counting processes and their inverses. Queueing Syst. 17 (1994) 107–128.
  • [11] T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York, 1985.
  • [12] T. M. Liggett. An improved subadditive ergodic theorem. Ann. Probab. 13 (1985) 1279–1285.
  • [13] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften 324. Springer, Berlin, 1999.
  • [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.
  • [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.
  • [16] J. van den Berg, O. Häggström and J. Kahn. Some conditional correlation inequalities for percolation and related processes. Random Structures Algorithms 29 (2006) 417–435.
  • [17] O. Zeitouni. Random walks in random environment. In XXXI Summer School in Probability, Saint-Flour, 2001 193–312. Lecture Notes in Math. 1837. Springer, Berlin, 2004.