Monotonicity of speed for leafless Galton–Watson trees

Determine whether the speed v(λ) of the λ-biased random walk on every supercritical Galton–Watson tree without leaves is nonincreasing on the full interval [0,m), where m is the mean offspring number.

Background

The paper studies the speed v(λ) of a λ-biased random walk on a Galton–Watson tree without leaves. Although monotonicity has been established for several small-bias ranges, the general conjecture remains unresolved. The paper proves strict decrease only for the specific offspring distribution uniform on {2,3}, and only up to λ=1.755, whereas the full ballistic interval for that law is [0,2.5).

References

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.

— A Computer-Assisted Proof of Speed Monotonicity for the Biased Random Walk on a Galton-Watson Tree Beyond the Known Range  (2609.29894 - Mandarapu et al., 24 Sep 2026) in Abstract; Section 1, Introduction

Near $\lambda=m$, the Einstein relation \citep{bhoz2013} identifies the slope at the endpoint; monotonicity near $m$ would follow if $v'$ were known to be continuous there, which is open \citep{benarousfribergh2016}.

— A Computer-Assisted Proof of Speed Monotonicity for the Biased Random Walk on a Galton-Watson Tree Beyond the Known Range  (2609.29894 - Mandarapu et al., 24 Sep 2026) in Section 2, Setting and known results