Scaling of a random walk on a supercritical contact process
den Hollander, F. ; dos Santos, R. S.
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 1276-1300 / Harvested from Numdam
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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.

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.

Publié le : 2014-01-01
DOI : https://doi.org/10.1214/13-AIHP561
Classification:  60F15,  60K35,  60K37,  82B41,  82C22,  82C44 
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     title = {Scaling of a random walk on a supercritical contact process},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {50},
     year = {2014},
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     doi = {10.1214/13-AIHP561},
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     language = {en},
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den Hollander, F.; dos Santos, R. S. Scaling of a random walk on a supercritical contact process. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) pp. 1276-1300. doi : 10.1214/13-AIHP561. http://gdmltest.u-ga.fr/item/AIHPB_2014__50_4_1276_0/

[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). | MR 2988390 | Zbl 1252.60097

[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. | MR 2664339 | Zbl 1198.82049

[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. | MR 2786643 | Zbl 1228.60113

[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]. | Zbl 1277.60185

[6] C. Bezuidenhout and G. Grimmett. Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19 (1991) 984-1009. | MR 1112404 | Zbl 0743.60107

[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. | MR 933991 | Zbl 0621.60108

[10] P. Glynn and W. Whitt. Large deviations behavior of counting processes and their inverses. Queueing Syst. 17 (1994) 107-128. | MR 1295409 | Zbl 0805.60023

[11] T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York, 1985. | MR 776231 | Zbl 0559.60078

[12] T. M. Liggett. An improved subadditive ergodic theorem. Ann. Probab. 13 (1985) 1279-1285. | MR 806224 | Zbl 0579.60023

[13] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften 324. Springer, Berlin, 1999. | MR 1717346 | Zbl 0949.60006

[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. | Numdam | MR 2199800 | Zbl 1087.60074

[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. | MR 1890289

[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. | MR 2268229 | Zbl 1112.60087

[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. | MR 2071631 | Zbl 1060.60103