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The Recurrence/Transience of Random Walks on a Bounded Grid in an Increasing Dimension

Authors Shuma Kumamoto, Shuji Kijima , Tomoyuki Shirai

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Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Random walk
  • dynamic graph
  • recurrence
  • transience
  • coupling

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Abstract

It is celebrated that a simple random walk on ℤ and ℤ² returns to the initial vertex v infinitely many times during infinitely many transitions, which is said recurrent, while it returns to v only finite times on ℤ^d for d ≥ 3, which is said transient. It is also known that a simple random walk on a growing region on ℤ^d can be recurrent depending on growing speed for any fixed d. This paper shows that a simple random walk on {0,1,…,N}ⁿ with an increasing n and a fixed N can be recurrent depending on the increasing speed of n. Precisely, we are concerned with a specific model of a random walk on a growing graph (RWoGG) and show a phase transition between the recurrence and transience of the random walk regarding the growth speed of the graph. For the proof, we develop a pausing coupling argument introducing the notion of weakly less homesick as graph growing (weakly LHaGG).

Cite As Get BibTex

Shuma Kumamoto, Shuji Kijima, and Tomoyuki Shirai. The Recurrence/Transience of Random Walks on a Bounded Grid in an Increasing Dimension. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.AofA.2024.22

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, JP23H01077, and JP23K25774.

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