## Electronic Journal of Probability

### Vertex reinforced non-backtracking random walks: an example of path formation

#### Abstract

This article studies vertex reinforced random walks that are non-backtracking (denoted VRNBW), i.e. U-turns forbidden. With this last property and for a strong reinforcement, the emergence of a path may occur with positive probability. These walks are thus useful to model the path formation phenomenon, observed for example in ant colonies. This study is carried out in two steps. First, a large class of reinforced random walks is introduced and results on the asymptotic behavior of these processes are proved. Second, these results are applied to VRNBWs on complete graphs and for reinforced weights $W(k)=k^\alpha$, with $\alpha \ge 1$. It is proved that for $\alpha >1$ and $3\le m< \frac{3\alpha -1} {\alpha -1}$, the walk localizes on $m$ vertices with positive probability, each of these $m$ vertices being asymptotically equally visited. Moreover the localization on $m>\frac{3\alpha -1} {\alpha -1}$ vertices is a.s. impossible.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 39, 38 pp.

Dates
Accepted: 10 April 2018
First available in Project Euclid: 9 May 2018

https://projecteuclid.org/euclid.ejp/1525852816

Digital Object Identifier
doi:10.1214/18-EJP167

Mathematical Reviews number (MathSciNet)
MR3806407

Zentralblatt MATH identifier
06868384

#### Citation

Le Goff, Line C.; Raimond, Olivier. Vertex reinforced non-backtracking random walks: an example of path formation. Electron. J. Probab. 23 (2018), paper no. 39, 38 pp. doi:10.1214/18-EJP167. https://projecteuclid.org/euclid.ejp/1525852816

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