## The Annals of Probability

### Localization on 4 sites for vertex-reinforced random walks on $\mathbb{Z}$

#### Abstract

We characterize nondecreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on $4$ sites. A phase transition appears for weights of order $n\log\log n$: for weights growing faster than this rate, the VRRW localizes almost surely on, at most, $4$ sites, whereas for weights growing slower, the VRRW cannot localize on less than $5$ sites. When $w$ is of order $n\log\log n$, the VRRW localizes almost surely on either $4$ or $5$ sites, both events happening with positive probability.

#### Article information

Source
Ann. Probab. Volume 42, Number 2 (2014), 527-558.

Dates
First available in Project Euclid: 24 February 2014

https://projecteuclid.org/euclid.aop/1393251295

Digital Object Identifier
doi:10.1214/12-AOP811

Mathematical Reviews number (MathSciNet)
MR3178466

Zentralblatt MATH identifier
1297.60062

#### Citation

Basdevant, Anne-Laure; Schapira, Bruno; Singh, Arvind. Localization on 4 sites for vertex-reinforced random walks on $\mathbb{Z}$. Ann. Probab. 42 (2014), no. 2, 527--558. doi:10.1214/12-AOP811. https://projecteuclid.org/euclid.aop/1393251295.

#### References

• [1] Basdevant, A. L., Schapira, B. and Singh, A. (2014). Localization of a vertex reinforced random walk on $\mathbb{Z}$ with sub-linear weight. Probab. Theory Related Fields. To appear.
• [2] Benaïm, M. and Tarrès, P. (2011). Dynamics of vertex-reinforced random walks. Ann. Probab. 39 2178–2223.
• [3] Chen, L. H. Y. (1978). A short note on the conditional Borel–Cantelli lemma. Ann. Probab. 6 699–700.
• [4] D. Coppersmith, D. and Diaconis, P. (1987). Random walk with reinforcement. Unpublished manuscript.
• [5] Davis, B. (1990). Reinforced random walk. Probab. Theory Related Fields 84 203–229.
• [6] Erschler, A., Tóth, B. and Werner, W. (2012). Some locally self-interacting walks on the integers. In Probability in Complex Physical Systems 313–338. Springer, Berlin.
• [7] Erschler, A., Tóth, B. and Werner, W. (2012). Stuck walks. Probab. Theory Related Fields 154 149–163.
• [8] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application: Probability and Mathematical Statistics. Academic Press, New York.
• [9] Pemantle, R. (1992). Vertex-reinforced random walk. Probab. Theory Related Fields 92 117–136.
• [10] Pemantle, R. (2007). A survey of random processes with reinforcement. Probab. Surv. 4 1–79.
• [11] Pemantle, R. and Volkov, S. (1999). Vertex-reinforced random walk on $\textbf{Z}$ has finite range. Ann. Probab. 27 1368–1388.
• [12] Schapira, B. (2012). A 0–1 law for vertex-reinforced random walks on $\mathbb{Z}$ with weight of order $k^{\alpha}$, $\alpha<1/2$. Electron. Commun. Probab. 17 1–8.
• [13] Sellke, T. (2008). Reinforced random walk on the $d$-dimensional integer lattice. Markov Process. Related Fields 14 291–308.
• [14] Tarrès, P. (2004). Vertex-reinforced random walk on $\mathbb{Z}$ eventually gets stuck on five points. Ann. Probab. 32 2650–2701.
• [15] Tarrès, P. (2011). Localization of reinforced random walks. Preprint. Available at arXiv:1103.5536.
• [16] Volkov, S. (2001). Vertex-reinforced random walk on arbitrary graphs. Ann. Probab. 29 66–91.
• [17] Volkov, S. (2006). Phase transition in vertex-reinforced random walks on $\mathbb{Z}$ with non-linear reinforcement. J. Theoret. Probab. 19 691–700.