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An Analysis of the Recurrence/Transience of Random Walks on Growing Trees and Hypercubes

Authors Shuma Kumamoto, Shuji Kijima , Tomoyuki Shirai

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Author Details

Shuma Kumamoto
  • Graduate School of Mathematical Science, Kyushu University, Fukuoka, Japan
Shuji Kijima
  • Faculty of Data Science, Shiga University, Hikone, Japan
Tomoyuki Shirai
  • Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

Funding

  • Kumamoto, Shuma: This work is supported by JST SPRING, Grant Number JPMJSP2136.
  • Kijima, Shuji: This work is partly supported by JSPS KAKENHI Grant Number JP21H03396.
  • Shirai, Tomoyuki: This work is partly supported by JSPS KAKENHI Grant Number JP20K20884, JP22H05105 and JP23H01077.

Cite AsGet BibTex

Shuma Kumamoto, Shuji Kijima, and Tomoyuki Shirai. An Analysis of the Recurrence/Transience of Random Walks on Growing Trees and Hypercubes. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SAND.2024.17

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Random walk
  • dynamic graph
  • recurrent
  • transient

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