Directed random walk on the backbone of an oriented percolation cluster

Matthias Birkner (Johannes-Gutenberg-Universität Mainz)
Jiri Cerny (University of Vienna)
Andrej Depperschmidt (Albert-Ludwigs-Universität Freiburg)
Nina Gantert (Technische Universität München)


We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the "ancestral lineage'' of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) using a regeneration approach. Furthermore, we obtain a quenched central limit theorem (i.e. for almost any realisation of the cluster) via an analysis of joint renewals of two independent walks on the same cluster.

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Pages: 1-35

Publication Date: August 31, 2013

DOI: 10.1214/EJP.v18-2302


  • 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 (2011), no. 21, 587--617. MR2786643
  • Bolthausen, Erwin; Sznitman, Alain-Sol. On the static and dynamic points of view for certain random walks in random environment. Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday. Methods Appl. Anal. 9 (2002), no. 3, 345--375. MR2023130
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Corrected reprint of the second (1998) edition. Stochastic Modelling and Applied Probability, 38. Springer-Verlag, Berlin, 2010. xvi+396 pp. ISBN: 978-3-642-03310-0 MR2571413
  • den Hollander, F.; dos Santos, R.; Sidoravicius, V. Law of large numbers for non-elliptic random walks in dynamic random environments. Stochastic Process. Appl. 123 (2013), no. 1, 156--190. MR2988114
  • Dolgopyat, Dmitry; Keller, Gerhard; Liverani, Carlangelo. Random walk in Markovian environment. Ann. Probab. 36 (2008), no. 5, 1676--1710. MR2440920
  • Dolgopyat, Dmitry; Liverani, Carlangelo. Non-perturbative approach to random walk in Markovian environment. Electron. Commun. Probab. 14 (2009), 245--251. MR2507753
  • Durrett, Richard. Oriented percolation in two dimensions. Ann. Probab. 12 (1984), no. 4, 999--1040. MR0757768
  • Etheridge, Alison. Some mathematical models from population genetics. Lectures from the 39th Probability Summer School held in Saint-Flour, 2009. Lecture Notes in Mathematics, 2012. Springer, Heidelberg, 2011. viii+119 pp. ISBN: 978-3-642-16631-0 MR2759587
  • Faddeev, D. K.; Faddeeva, V. N. Computational methods of linear algebra. Translated by Robert C. Williams W. H. Freeman and Co., San Francisco-London 1963 xi+621 pp. MR0158519
  • Grimmett, Geoffrey; Hiemer, Philipp. Directed percolation and random walk. In and out of equilibrium (Mambucaba, 2000), 273--297, Progr. Probab., 51, Birkhäuser Boston, Boston, MA, 2002. MR1901958
  • Gut, Allan. Stopped random walks. Limit theorems and applications. Applied Probability. A Series of the Applied Probability Trust, 5. Springer-Verlag, New York, 1988. x+199 pp. ISBN: 0-387-96590-4 MR0916870
  • Hall, P.; Heyde, C. C. Martingale limit theory and its application. Probability and Mathematical Statistics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. xii+308 pp. ISBN: 0-12-319350-8 MR0624435
  • Joseph, Mathew; Rassoul-Agha, Firas. Almost sure invariance principle for continuous-space random walk in dynamic random environment. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 43--57. MR2748407
  • Kuczek, Thomas. The central limit theorem for the right edge of supercritical oriented percolation. Ann. Probab. 17 (1989), no. 4, 1322--1332. MR1048929
  • Liggett, Thomas M. Stochastic interacting systems: contact, voter and exclusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 324. Springer-Verlag, Berlin, 1999. xii+332 pp. ISBN: 3-540-65995-1 MR1717346
  • P. Mörters and Y. Peres, Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010, With an appendix by Oded Schramm and Wendelin Werner. MR2604525 (2011i:60152)
  • Neuhauser, Claudia. Ergodic theorems for the multitype contact process. Probab. Theory Related Fields 91 (1992), no. 3-4, 467--506. MR1151806
  • Račkauskas, Alfredas. On the rate of convergence in the martingale CLT. Statist. Probab. Lett. 23 (1995), no. 3, 221--226. MR1340154
  • F. Redig and Völlering F., Limit theorems for random walks in dynamic random environment, To appear in Ann. Probab (2013).
  • A. Sarkar and R. Sun, Brownian web in the scaling limit of supercritical oriented percolation in dimension 1 + 1, Electron. J. Probab. 18 (2013), no. 21, 1--23.
  • Stewart, G. W.; Sun, Ji Guang. Matrix perturbation theory. Computer Science and Scientific Computing. Academic Press, Inc., Boston, MA, 1990. xvi+365 pp. ISBN: 0-12-670230-6 MR1061154
  • Sznitman, Alain-Sol. Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. (JEMS) 2 (2000), no. 2, 93--143. MR1763302
  • Valesin, Daniel. Multitype contact process on $\Bbb Z$: extinction and interface. Electron. J. Probab. 15 (2010), no. 73, 2220--2260. MR2748404

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