发表机构
VaidhyaMegha Private Limited（VaidhyaMegha私人有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文通过计算机辅助证明，对后代分布为{2,3}的Galton-Watson树上偏置随机游走，在偏置范围[0,1.755]内速度严格递减，扩展了已知的单调性范围。
AI 中文摘要
对于无叶子的上临界Galton-Watson树上λ-偏置随机游走的速度v(λ)，猜想其在[0,m)上非增，其中m为平均后代数。单调性仅在较小偏置下已知：λ≤1/1160，λ≤1/2，并且当每个顶点至少有m_1≥2个孩子时，λ≤m_1/(1+sqrt(1-1/m_1))。对于后代均匀分布在{2,3}（m=2.5）的情况，最后一个界限为1.1716。我们通过计算机辅助证明，对于该分布，v在[0,1.755]上严格递减。证明分为三个部分。Aidekon的速度公式给出v=(R-λ)/(R+λ)，其中R为显式泛函，因此v递减当且仅当R/λ递减；我们直接在两个偏置下比较R/λ，这避免了微分电导。电导关于λ的逐路径Lipschitz界将该比较转化为显式函数期望之间的不等式。离散化律的单调夹逼给出电导律的两侧界，每个λ-单元在浮点误差有界的基础上用精确有理算术验证；独立的区间算术实现在抽查单元上一致。该方法在粗略Lipschitz界变得过弱处停止；对电导导数的更精细控制是全范围所需。代码和证书已公开。
英文摘要
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.
Comments9 pages, 1 figure. Computer-assisted proof; one command reproduces every certificate. Code: https://github.com/samyama-ai/gw-speed-certificate. v2: corrects the quoted Song-Wang-Xiang bound to 2/(1+sqrt(1/2))=1.17157..., adds underflow to the floating-point bound, cites Song-Liu (2026), and re-checks the endpoint cell in interval arithmetic
