A Computer-Assisted Proof of Speed Monotonicity for the Biased Random Walk on a Galton-Watson Tree Beyond the Known Range
Abstract
The speed v(lambda) of the lambda-biased random walk on a supercritical Galton-Watson tree without leaves is conjectured to be nonincreasing on [0,m), where m is the mean offspring. For non-constant offspring, monotonicity is known only for small bias: lambda <= 1/1160, lambda <= 1/2, and, when every vertex has at least m_1 >= 2 children, lambda <= m_1/(1+sqrt(1-1/m_1)). For offspring uniform on {2,3} (m=2.5) the last bound is 2/(1+sqrt(1/2)) = 1.17157... We prove, with computer assistance, that v is strictly decreasing on [0,1.755] for this law. The proof has three parts. Aidekon's speed formula gives v=(R-lambda)/(R+lambda) for an explicit functional R, so v decreases exactly when R/lambda does; we compare R/lambda at two biases directly, which avoids differentiating the conductance. A pathwise Lipschitz bound on the conductance in lambda turns that comparison into an inequality between expectations of explicit functions. A monotone sandwich of discretised laws gives two-sided bounds on the conductance law, and each lambda-cell is verified with exact rational arithmetic on top of bounded floating-point error; an independent interval-arithmetic implementation re-certifies five cells, including the one that sets the endpoint. The method stops where the crude Lipschitz bound becomes too weak; sharper control of the derivative of the conductance is what the full range needs. Code and certificates are public.
Related
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- On non-monotonicity of logarithmic energy for random matrices
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BibTeX
@misc{mandarapu2026computerassisted,
title = {{A Computer-Assisted Proof of Speed Monotonicity for the Biased Random Walk on a Galton-Watson Tree Beyond the Known Range}},
author = {Madhulatha Mandarapu and Sandeep Kunkunuru},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29894},
eprint = {2609.29894},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29894},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Computer-Assisted Proof of Speed Monotonicity for the Biased Random Walk on a Galton-Watson Tree Beyond the Known Range
AU - Madhulatha Mandarapu
AU - Sandeep Kunkunuru
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.29894
UR - https://arxiv.org/abs/2609.29894
AB - The speed v(lambda) of the lambda-biased random walk on a supercritical Galton-Watson tree without leaves is conjectured to be nonincreasing on [0,m), where m is the mean offspring. For non-constant offspring, monotonicity is known only for small bias: lambda <= 1/1160, lambda <= 1/2, and, when every vertex has at least m_1 >= 2 children, lambda <= m_1/(1+sqrt(1-1/m_1)). For offspring uniform on {2,3} (m=2.5) the last bound is 2/(1+sqrt(1/2)) = 1.17157... We prove, with computer assistance, that v is strictly decreasing on [0,1.755] for this law. The proof has three parts. Aidekon's speed formula gives v=(R-lambda)/(R+lambda) for an explicit functional R, so v decreases exactly when R/lambda does; we compare R/lambda at two biases directly, which avoids differentiating the conductance. A pathwise Lipschitz bound on the conductance in lambda turns that comparison into an inequality between expectations of explicit functions. A monotone sandwich of discretised laws gives two-sided bounds on the conductance law, and each lambda-cell is verified with exact rational arithmetic on top of bounded floating-point error; an independent interval-arithmetic implementation re-certifies five cells, including the one that sets the endpoint. The method stops where the crude Lipschitz bound becomes too weak; sharper control of the derivative of the conductance is what the full range needs. Code and certificates are public.
ER -