eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-05-31
17:1
17:17
10.4230/LIPIcs.SAND.2024.17
article
An Analysis of the Recurrence/Transience of Random Walks on Growing Trees and Hypercubes
Kumamoto, Shuma
1
Kijima, Shuji
2
https://orcid.org/0000-0001-6061-2330
Shirai, Tomoyuki
3
https://orcid.org/0000-0001-6269-5387
Graduate School of Mathematical Science, Kyushu University, Fukuoka, Japan
Faculty of Data Science, Shiga University, Hikone, Japan
Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan
It is a celebrated fact that a simple random walk on an infinite k-ary tree for k ≥ 2 returns to the initial vertex at most finitely many times during infinitely many transitions; it is called transient. This work points out the fact that a simple random walk on an infinitely growing k-ary tree can return to the initial vertex infinitely many times, it is called recurrent, depending on the growing speed of the tree. Precisely, this paper is concerned with a simple specific model of a random walk on a growing graph (RWoGG), and shows a phase transition between the recurrence and transience of the random walk regarding the growing speed of the graph. To prove the phase transition, we develop a coupling argument, introducing the notion of less homesick as graph growing (LHaGG). We also show some other examples, including a random walk on {0,1}ⁿ with infinitely growing n, of the phase transition between the recurrence and transience. We remark that some graphs concerned in this paper have infinitely growing degrees.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol292-sand2024/LIPIcs.SAND.2024.17/LIPIcs.SAND.2024.17.pdf
Random walk
dynamic graph
recurrent
transient