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

Authors Shuma Kumamoto, Shuji Kijima , Tomoyuki Shirai



PDF
Thumbnail PDF

File

LIPIcs.AofA.2024.22.pdf
  • Filesize: 0.7 MB
  • 15 pages

Document Identifiers

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

Cite AsGet 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

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).

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Gideon Amir, Itai Benjamini, Ori Gurel-Gurevich, and Gady Kozma. Random walk in changing environment. Stochastic Processes and their Applications, 130(12):7463-7482, 2020. URL: https://doi.org/10.1016/j.spa.2020.08.003.
  2. John Augustine, Gopal Pandurangan, and Peter Robinson. Distributed algorithmic foundations of dynamic networks. SIGACT News, 47(1):69-98, March 2016. URL: https://doi.org/10.1145/2902945.2902959.
  3. Chen Avin, Michal Koucký, and Zvi Lotker. How to explore a fast-changing world (cover time of a simple random walk on evolving graphs). In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming, ICALP 2008, pages 121-132, 2008. Google Scholar
  4. Chen Avin, Michal Koucký, and Zvi Lotker. Cover time and mixing time of random walks on dynamic graphs. Random Structures & Algorithms, 52(4):576-596, 2018. URL: https://doi.org/10.1002/rsa.20752.
  5. Itai Benjamini and David Wilson. Excited Random Walk. Electronic Communications in Probability, 8:86-92, 2003. URL: https://doi.org/10.1214/ECP.v8-1072.
  6. Leran Cai, Thomas Sauerwald, and Luca Zanetti. Random walks on randomly evolving graphs. In Andrea Werneck Richa and Christian Scheideler, editors, Structural Information and Communication Complexity, SIROCCO 2020, pages 111-128, 2020. Google Scholar
  7. Colin Cooper. Random walks, interacting particles, dynamic networks: Randomness can be helpful. In Adrian Kosowski and Masafumi Yamashita, editors, Structural Information and Communication Complexity, SIROCCO 2011, pages 1-14, 2011. Google Scholar
  8. Colin Cooper and Alan Frieze. Crawling on simple models of web graphs. Internet Mathematics, 1(1):57-90, 2003. Google Scholar
  9. Burgess Davis. Reinforced random walk. Probability Theory and Related Fields, 84(2):203-229, 1990. URL: https://doi.org/10.1007/BF01197845.
  10. Amir Dembo, Ruojun Huang, Ben Morris, and Yuval Peres. Transience in growing subgraphs via evolving sets. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 53(3):1164-1180, 2017. URL: https://doi.org/10.1214/16-AIHP751.
  11. Amir Dembo, Ruojun Huang, and Vladas Sidoravicius. Monotone interaction of walk and graph: recurrence versus transience. Electronic Communications in Probability, 19:1-12, 2014. URL: https://doi.org/10.1214/ECP.v19-3607.
  12. Amir Dembo, Ruojun Huang, and Vladas Sidoravicius. Walking within growing domains: recurrence versus transience. Electronic Journal of Probability, 19:1-20, 2014. URL: https://doi.org/10.1214/EJP.v19-3272.
  13. Amir Dembo, Ruojun Huang, and Tianyi Zheng. Random walks among time increasing conductances: heat kernel estimates. Probability Theory and Related Fields, 175(1):397-445, October 2019. URL: https://doi.org/10.1007/s00440-018-0894-1.
  14. Oksana Denysyuk and Luís Rodrigues. Random walks on evolving graphs with recurring topologies. In Fabian Kuhn, editor, Distributed Computing, DISC 2014, pages 333-345, 2014. Google Scholar
  15. Dmitry Dolgopyat, Gerhard Keller, and Carlangelo Liverani. Random walk in Markovian environment. The Annals of Probability, 36(5):1676-1710, 2008. URL: https://doi.org/10.1214/07-AOP369.
  16. Richard Durrett. Probability: theory and examples. Cambridge University Press, Fifth edition, 2019. Google Scholar
  17. Daniel Figueiredo, Giulio Iacobelli, Roberto Oliveira, Bruce Reed, and Rodrigo Ribeiro. On a random walk that grows its own tree. Electronic Journal of Probability, 26:1-40, 2021. URL: https://doi.org/10.1214/20-EJP574.
  18. Ruojun Huang. On random walk on growing graphs. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 55(2):1149-1162, 2019. URL: https://doi.org/10.1214/18-AIHP913.
  19. Giulio Iacobelli, Daniel R. Figueiredo, and Giovanni Neglia. Transient and slim versus recurrent and fat: Random walks and the trees they grow. Journal of Applied Probability, 56(3):769-786, 2019. URL: https://doi.org/10.1017/jpr.2019.43.
  20. Shuji Kijima, Nobutaka Shimizu, and Takeharu Shiraga. How many vertices does a random walk miss in a network with moderately increasing the number of vertices? In Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 106-122, 2021. URL: https://doi.org/10.1137/1.9781611976465.8.
  21. Elena Kosygina and Martin PW Zerner. Excited random walks: results, methods, open problems. Bulletin of the Institute of Mathematics Academia Sinica (New Series), 8(1), 2013. Google Scholar
  22. Fabian Kuhn and Rotem Oshman. Dynamic networks: models and algorithms. SIGACT News, 42(1):82-96, March 2011. URL: https://doi.org/10.1145/1959045.1959064.
  23. Shuma Kumamoto, Shuji Kijima, and Tomoyuki Shirai. An analysis of the recurrence/transience of random walks on growing trees and hypercubes, 2024. URL: https://arxiv.org/abs/2405.09102.
  24. Ioannis Lamprou, Russell Martin, and Paul Spirakis. Cover time in edge-uniform stochastically-evolving graphs. Algorithms, 11(10), 2018. URL: https://doi.org/10.3390/a11100149.
  25. David A. Levin and Yuval Peres. Markov chains and mixing times. American Mathematical Society, Second edition, 2017. Google Scholar
  26. Russell Lyons. Random walks and percolation on trees. The Annals of Probability, 18(3):931-958, 1990. Google Scholar
  27. Russell Lyons and Yuval Peres. Probability on Trees and Networks. Cambridge University Press, USA, 2017. Google Scholar
  28. Othon Michail and Paul G. Spirakis. Elements of the theory of dynamic networks. Communications of the ACM, 61(2):72, January 2018. URL: https://doi.org/10.1145/3156693.
  29. Laurent Saloff-Coste and Jessica Zúñiga. Merging for time inhomogeneous finite Markov chains, Part I: Singular values and stability. Electronic Journal of Probability, 14:1456-1494, 2009. URL: https://doi.org/10.1214/EJP.v14-656.
  30. Laurent Saloff-Coste and Jessica Zúñiga. Merging for time inhomogeneous finite Markov chains, Part II: Nash and log-Sobolev inequalities. The Annals of Probability, 39(3):1161-1203, 2011. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail