Electronic Journal of Probability

On the speed of once-reinforced biased random walk on trees

Andrea Collevecchio, Mark Holmes, and Daniel Kious

Full-text: Open access

Abstract

We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative once-reinforcement the ORbRW can be recurrent even when the underlying biased random walk is ballistic. We also prove that, on Galton-Watson trees without leaves, the speed is positive in the transient regime. Finally, we prove that, on regular trees, the speed of the ORbRW is monotone decreasing in the reinforcement parameter when the underlying random walk has high speed, and the reinforcement parameter is small.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 86, 32 pp.

Dates
Received: 6 September 2017
Accepted: 27 July 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1536717745

Digital Object Identifier
doi:10.1214/18-EJP208

Mathematical Reviews number (MathSciNet)
MR3858914

Zentralblatt MATH identifier
06964780

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
random walk once-reinforced random walk Galton-Watson tree reinforcement

Rights
Creative Commons Attribution 4.0 International License.

Citation

Collevecchio, Andrea; Holmes, Mark; Kious, Daniel. On the speed of once-reinforced biased random walk on trees. Electron. J. Probab. 23 (2018), paper no. 86, 32 pp. doi:10.1214/18-EJP208. https://projecteuclid.org/euclid.ejp/1536717745


Export citation

References

  • [1] Aidekon E. (2014) Speed of the biased random walk on a Galton–Watson tree. Probability Theory and Related Fields 159, 597–617.
  • [2] Angel, O., Crawford, N., Kozma, G. (2014) Localization for linearly edge reinforced random walks. Duke Math. J. 163(5):889–921.
  • [3] Barbour, A, Collevecchio, A. (2017) General random walk in a random environment defined on Galton–Watson trees. Forthcoming Annales de L’Institut Henri Poincaré.
  • [4] Ben Arous, G. and Fribergh, A. and Sidoravicius, V. (2014) Lyons-Pemantle-Peres monotonicity problem for high biases. Comm. Pure Appl. Math. 167(4), 519–530.
  • [5] Collevecchio, A. (2006) On the transience of processes defined on Galton-Watson trees. Ann. Probab. 34, 870–878.
  • [6] Coppersmith, D., and Diaconis, P. (1987) Reinforced random walk, Unpublished manuscript.
  • [7] Davis, B. (1990) Reinforced random walk, Probability Theory and Related Fields 84, 203–229.
  • [8] Disertori, M., Sabot, C., and Tarrès, P. (2015) Transience of Edge-Reinforced Random Walk Communications in Mathematical Physics, 339(1):121–148
  • [9] Durrett, R., Kesten, H., and Limic, V., (2002) Once edge-reinforced random walk on a tree, Probability Theory and Related Fields 122(4), 567–592.
  • [10] Hofstad, R. van der and Holmes, M. (2010) A monotonicity property for excited random walk in high dimensions. Prob. Theory Relat. Fields. 147:333–348.
  • [11] Hofstad, R. van der and Holmes, M. (2012) An expansion for self-interacting random walks. Brazilian Journal of Probability and Statistics 26:1–55.
  • [12] Holmes, M. (2012) Excited against the tide: A random walk with competing drifts. Annales de l’Institut Henri Poincare Probab. Statist. 48:745–773.
  • [13] Holmes, M. (2015) On strict monotonicity of the speed for excited random walks in one dimension. Electron. Commun. Probab. 20, no. 41, 1–7.
  • [14] Holmes, M. and Salisbury, T.S. (2012) A combinatorial result with applications to self-interacting random walks. Journal of Combinatorial Theory, Series A 119:460–475.
  • [15] Holmes, M. and Salisbury, T.S. (2014) Random walks in degenerate random environments. Canad. J. Math. 66:1050–1077.
  • [16] Holmes, M. and Sun, R. (2012) A monotonicity property for random walk in a partially random environment. Stochastic Processes and their Applications 122:1369–1396.
  • [17] Hummel, P.M. (1940). A note on Stirling’s formula. Amer. Math. Month., 47 (2), 97–99.
  • [18] Kious, D., Sidoravicius, V. (2018) Phase transition for the Once-reinforced random walk on $\mathbb{Z} ^d$-like trees. Ann. Probab. 46, 2121–2133.
  • [19] Lyons, R. and Pemantle, R. and Peres, Y. (1996). Biased random walks on Galton-Watson treess. Probab. Theory Related Fields 106(2), 249–264.
  • [20] Pemantle, R. (1988). Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16, 1229–1241.
  • [21] Pham, C.D. (2015) Monotonicity and regularity of the speed for excited random walks in higher dimensions. Electronic J. of Probability Volume 20, paper no. 72
  • [22] Pham, C.D. Some results on regularity and monotonicity of the speed for excited random walk in low dimensions. to appear in Stoch. Proc. Appl.
  • [23] Sabot, C. and Tarrès, P. (2015) Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. 17(9), 2353–2378.
  • [24] Sabot, C. and Zeng, X. (2015) A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs. preprint, arXiv:1507.07944.
  • [25] Zeitouni, O. (2004) Random Walks in Random Environment, XXXI summer school in probability, St Flour (2001), Lecture Notes in Math. 1837, pp. 193–312. Springer, Berlin.