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Documents authored by Shirai, Tomoyuki

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DOI: 10.4230/LIPIcs.AofA.2024.22
The Recurrence/Transience of Random Walks on a Bounded Grid in an Increasing Dimension

Authors: Shuma Kumamoto, Shuji Kijima, and Tomoyuki Shirai

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

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).
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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)

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@InProceedings{kumamoto_et_al:LIPIcs.AofA.2024.22,
  author =	{Kumamoto, Shuma and Kijima, Shuji and Shirai, Tomoyuki},
  title =	{{The Recurrence/Transience of Random Walks on a Bounded Grid in an Increasing Dimension}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{22:1--22:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.22},
  URN =		{urn:nbn:de:0030-drops-204577},
  doi =		{10.4230/LIPIcs.AofA.2024.22},
  annote =	{Keywords: Random walk, dynamic graph, recurrence, transience, coupling}
}
Document
DOI: 10.4230/LIPIcs.SAND.2024.17
An Analysis of the Recurrence/Transience of Random Walks on Growing Trees and Hypercubes

Authors: Shuma Kumamoto, Shuji Kijima, and Tomoyuki Shirai

Published in: LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

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.
Cite as

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)

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@InProceedings{kumamoto_et_al:LIPIcs.SAND.2024.17,
  author =	{Kumamoto, Shuma and Kijima, Shuji and Shirai, Tomoyuki},
  title =	{{An Analysis of the Recurrence/Transience of Random Walks on Growing Trees and Hypercubes}},
  booktitle =	{3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)},
  pages =	{17:1--17:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-315-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{292},
  editor =	{Casteigts, Arnaud and Kuhn, Fabian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.17},
  URN =		{urn:nbn:de:0030-drops-198955},
  doi =		{10.4230/LIPIcs.SAND.2024.17},
  annote =	{Keywords: Random walk, dynamic graph, recurrent, transient}
}
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