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.
Citation
Anne-Laure Basdevant. Bruno Schapira. Arvind Singh. "Localization on 4 sites for vertex-reinforced random walks on $\mathbb{Z}$." Ann. Probab. 42 (2) 527 - 558, March 2014. https://doi.org/10.1214/12-AOP811
Information