Spectral radius of biased random walks on regular trees

He Song; Meng Liu; He Song; Meng Liu

doi:10.3934/math.2026195

AIMS Mathematics

2026, Volume 11, Issue 2: 4787-4804. doi: 10.3934/math.2026195

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Research article

Spectral radius of biased random walks on regular trees

He Song ,
Meng Liu ^,

School of Mathematics and Statistics, Huaiyin Normal University, Huai'an 223300, China

Received: 23 November 2025 Revised: 27 January 2026 Accepted: 03 February 2026 Published: 27 February 2026
MSC : Primary 05C81, 60G50, 60J10; Secondary 05C63, 05C80, 60C05

For the biased random walk on $ d $-regular trees, we obtain the spectral radius, first return probability and $ n $-step transition probability, speed. We prove that the spectral radius depends continuously on the bias parameter and is strictly increasing with respect to it, while the speed remains strictly positive and decreases monotonically as the bias increases.
- biased random walk,
- spectral radius,
- continuous
Citation: He Song, Meng Liu. Spectral radius of biased random walks on regular trees[J]. AIMS Mathematics, 2026, 11(2): 4787-4804. doi: 10.3934/math.2026195

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Abstract

For the biased random walk on $ d $-regular trees, we obtain the spectral radius, first return probability and $ n $-step transition probability, speed. We prove that the spectral radius depends continuously on the bias parameter and is strictly increasing with respect to it, while the speed remains strictly positive and decreases monotonically as the bias increases.

References

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