Electronic Journal of Probability

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

Line C. Le Goff and Olivier Raimond

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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
Received: 31 July 2017
Accepted: 10 April 2018
First available in Project Euclid: 9 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1525852816

Digital Object Identifier
doi:10.1214/18-EJP167

Mathematical Reviews number (MathSciNet)
MR3806407

Zentralblatt MATH identifier
06868384

Subjects
Primary: 60G99: None of the above, but in this section 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 91C99: None of the above, but in this section

Keywords
vertex reinforced random walk non-backtracking random walks complete graph path formation

Rights
Creative Commons Attribution 4.0 International License.

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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