A Computer-Assisted Proof of Speed Monotonicity for the Biased Random Walk on a Galton-Watson Tree Beyond the Known Range

Source 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. Monotonicity is known only for small bias: lambda = 2 children, lambda <= m_1/(1+sqrt(1-1/m_1)). For offspring uniform on {2,3} (m=2.5) the last bound is 1.1716. 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 agrees on spot cells. 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.