The Annals of Probability

Phase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like trees

Daniel Kious and Vladas Sidoravicius

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this short paper, we consider the Once-reinforced random walk with reinforcement parameter $a$ on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter $a_{0}$ such that the Once-reinforced random walk is almost surely recurrent if $a>a_{0}$ and almost surely transient if $a<a_{0}$. This provides the first examples of phase transition for the Once-reinforced random walk.

Article information

Source
Ann. Probab., Volume 46, Number 4 (2018), 2121-2133.

Dates
Received: May 2016
Revised: June 2017
First available in Project Euclid: 13 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1528876823

Digital Object Identifier
doi:10.1214/17-AOP1222

Mathematical Reviews number (MathSciNet)
MR3813987

Zentralblatt MATH identifier
06919020

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

Keywords
Once-reinforced random walk recurrence transience phase transition

Citation

Kious, Daniel; Sidoravicius, Vladas. Phase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like trees. Ann. Probab. 46 (2018), no. 4, 2121--2133. doi:10.1214/17-AOP1222. https://projecteuclid.org/euclid.aop/1528876823


Export citation

References

  • [1] Agresti, A. (1975). On the extinction times of varying and random environment branching processes. J. Appl. Probab. 12 39–46.
  • [2] Angel, O., Crawford, N. and Kozma, G. (2014). Localization for linearly edge reinforced random walks. Duke Math. J. 163 889–921.
  • [3] Collevecchio, A. (2006). On the transience of processes defined on Galton–Watson trees. Ann. Probab. 34 870–878.
  • [4] Coppersmith, D. and Diaconis, P. (1986). Random walks with reinforcement. Unpublished manuscript.
  • [5] Davis, B. (1990). Reinforced random walk. Probab. Theory Related Fields 84 203–229.
  • [6] Disertori, M., Sabot, C. and Tarrès, P. (2015). Transience of edge-reinforced random walk. Comm. Math. Phys. 339 121–148.
  • [7] Disertori, M., Spencer, T. and Zirnbauer, M. R. (2010). Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model. Comm. Math. Phys. 300 435–486.
  • [8] Doyle, P. G. and Snell, E. J. (1984). Random Walks and Electrical Networks. Carus Math. Monographs 22. Math. Assoc. Amer, Washington, DC.
  • [9] Durrett, R., Kesten, H. and Limic, V. (2002). Once edge-reinforced random walk on a tree. Probab. Theory Related Fields 122 567–592.
  • [10] Kozma, G. (2012). Reinforced random walks. Preprint. arXiv:1208.0364.
  • [11] Lyons, R. and Pemantle, R. (1992). Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 125–136.
  • [12] Lyons, R. and Peres, Y. (2016). Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics 42. Cambridge Univ. Press. Available at http://pages.iu.edu/~rdlyons/.
  • [13] Merkl, F. and Rolles, S. W. W. (2009). Recurrence of edge-reinforced random walk on a two-dimensional graph. Ann. Probab. 37 1679–1714.
  • [14] Pemantle, R. (1988). Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 1229–1241.
  • [15] Pemantle, R. (2007). A survey of random processes with reinforcement. Probab. Surv. 4 1–79.
  • [16] Sabot, C. and Tarrès, P. (2015). Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. (JEMS) 17 2353–2378.
  • [17] Sabot, C. and Tarrès, P. (2016). Inverting Ray–Knight identity. Probab. Theory Related Fields. 165 559–580.
  • [18] Sabot, C., Tarrès, P. and Zeng, X. (2015). The vertex reinforced jump process and a random Schrödinger operator on finite graphs. Preprint. arXiv:1507.04660.
  • [19] Sabot, C. and Zeng, X. (2015). A random Schrödinger operator associated with the vertex reinforced jump process on infinite graphs. preprint, arXiv:1507.07944.
  • [20] Sellke, T. (2006). Recurrence of reinforced random walk on a ladder. Electron. J. Probab. 11 301–310.
  • [21] Tarrès, P. (2011). Localization of reinforced random walks. Preprint. arXiv:1103.5536.
  • [22] Vervoort, M. (2002). Reinforced random walks. Unpublished manuscript.