## Electronic Journal of Probability

### The speed of critically biased random walk in a one-dimensional percolation model

#### Abstract

We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and Häggström and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z} ^d$, namely, for some critical value $\lambda _{\mathrm{c} }>0$ of the bias, it holds that the asymptotic linear speed $\overline{\mathrm {v}}$ of the walk is strictly positive if the bias $\lambda$ is strictly smaller than $\lambda _{\mathrm{c} }$, whereas $\overline{\mathrm {v}} =0$ if $\lambda \geq \lambda _{\mathrm{c} }$.

We show that at the critical bias $\lambda = \lambda _{\mathrm{c} }$, the displacement of the random walk from the origin is of order $n/\log n$. This is in accordance with simulation results by Dhar and Stauffer for biased random walk on the infinite cluster of supercritical bond percolation on $\mathbb{Z} ^d$.

Our result is based on fine estimates for the tails of suitable regeneration times. As a by-product of these estimates we also obtain the order of fluctuations of the walk in the sub-ballistic and in the ballistic, nondiffusive phase.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 23, 29 pp.

Dates
Accepted: 9 February 2019
First available in Project Euclid: 23 March 2019

https://projecteuclid.org/euclid.ejp/1553306439

Digital Object Identifier
doi:10.1214/19-EJP277

#### Citation

Lübbers, Jan-Erik; Meiners, Matthias. The speed of critically biased random walk in a one-dimensional percolation model. Electron. J. Probab. 24 (2019), paper no. 23, 29 pp. doi:10.1214/19-EJP277. https://projecteuclid.org/euclid.ejp/1553306439

#### References

• [1] Marina Axelson-Fisk and Olle Häggström, Biased random walk in a one-dimensional percolation model, Stochastic Process. Appl. 119 (2009), no. 10, 3395–3415.
• [2] Marina Axelson-Fisk and Olle Häggström, Conditional percolation on one-dimensional lattices, Adv. in Appl. Probab. 41 (2009), no. 4, 1102–1122.
• [3] Mustansir Barma and Deepak Dhar, Directed diffusion in a percolation network, Journal of Physics C: Solid State Physics 16 (1983), no. 8, 1451.
• [4] Gérard Ben Arous and Alexander Fribergh, Biased random walks on random graphs, Probability and statistical physics in St. Petersburg, Proc. Sympos. Pure Math., vol. 91, Amer. Math. Soc., Providence, RI, 2016, pp. 99–153.
• [5] Gérard Ben Arous, Alexander Fribergh, Nina Gantert, and Alan Hammond, Biased random walks on Galton-Watson trees with leaves, Ann. Probab. 40 (2012), no. 1, 280–338.
• [6] Noam Berger, Nina Gantert, and Yuval Peres, The speed of biased random walk on percolation clusters, Probab. Theory Related Fields 126 (2003), no. 2, 221–242.
• [7] N. H. Bingham, Limit theorems for regenerative phenomena, recurrent events and renewal theory, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 21 (1972), 20–44.
• [8] Nicholas H. Bingham, Charles M. Goldie, and Józef L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1989.
• [9] Deepak Dhar and Dietrich Stauffer, Drift and trapping in biased diffusion on disordered lattices, International Journal of Modern Physics C 09 (1998), no. 02, 349–355.
• [10] Nathanaël Enriquez, Christophe Sabot, and Olivier Zindy, Limit laws for transient random walks in random environment on $\mathbb{Z}$, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2469–2508.
• [11] William Feller, An Introduction to Probability Theory and its Applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966.
• [12] Alexander Fribergh and Alan Hammond, Phase transition for the speed of the biased random walk on the supercritical percolation cluster, Comm. Pure Appl. Math. 67 (2014), no. 2, 173–245.
• [13] Alexander Fribergh and Daniel Kious, Scaling limits for sub-ballistic biased random walks in random conductances, Ann. Probab. 46 (2018), no. 2, 605–686.
• [14] Nina Gantert, Matthias Meiners, and Sebastian Müller, Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model, J. Stat. Phys. 170 (2018), no. 6, 1123–1160.
• [15] Allan Gut, Stopped Random Walks, second ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2009, Limit theorems and applications.
• [16] I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971, With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman.
• [17] Alexander Iksanov, Alexander Marynych, and Matthias Meiners, Moment convergence of first-passage times in renewal theory, Statist. Probab. Lett. 119 (2016), 134–143.
• [18] H. Kesten, M. V. Kozlov, and F. Spitzer, A limit law for random walk in a random environment, Compositio Math. 30 (1975), 145–168.
• [19] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, Providence, RI, 2009, With a chapter by James G. Propp and David B. Wilson.
• [20] Russell Lyons, Robin Pemantle, and Yuval Peres, Biased random walks on Galton-Watson trees, Probab. Theory Related Fields 106 (1996), no. 2, 249–264.
• [21] Eddy Mayer-Wolf, Alexander Roitershtein, and Ofer Zeitouni, Limit theorems for one-dimensional transient random walks in Markov environments, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 5, 635–659.
• [22] Alain-Sol Sznitman, On the anisotropic walk on the supercritical percolation cluster, Comm. Math. Phys. 240 (2003), no. 1-2, 123–148.
• [23] Ofer Zeitouni, Random walks in random environment, Lecture notes on probability theory and statistics, Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004, pp. 189–312.