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The speed of a biased walk on a Galton–Watson tree without leaves is monotonic for low values of bias

Part of: Markov processes

Published online by Cambridge University Press:  05 March 2025

He Song ,
Longmin Wang  and
Kainan Xiang
He Song*
Affiliation:
Huaiyin Normal University
Longmin Wang*
Affiliation:
Nankai University
Kainan Xiang*
Affiliation:
Xiangtan University
*
*Postal address: School of Mathematics and Statistics, Huaiyin Normal University Huaiyin 223300, P. R. China. Email: songhe@hytc.edu.cn
**Postal address: School of Statistics and Data Science, Nankai University Tianjin 300071, P. R. China. Email: wanglm@nankai.edu.cn
***Postal address: School of Mathematics and Computational Science, Xiangtan University Xiangtan City 210000, Hunan Province, P. R. China. Email: kainan.xiang@xtu.edu.cn
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Abstract

We show that for $\lambda\in[0,{m_1}/({1+\sqrt{1-{1}/{m_1}}})]$, the biased random walk’s speed on a Galton–Watson tree without leaves is strictly decreasing, where $m_1\geq 2$. Our result extends the monotonic interval of the speed on a Galton–Watson tree.

Keywords

Galton–Watson treebiased random walkspeedmonotonicity

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 3 , September 2025 , pp. 1044 - 1052
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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