Differentiability and monotonicity of expected passage time in Euclidean first-passage percolation

In first-passage percolation (FPP) models, the passage time T ℓ from the origin to the point ℓe ℓ satisfies f(ℓ) := ET ℓ = μℓ + o(ℓ ½+ε), where μ ∊ (0,∞) denotes the time constant. Yet, for lattice FPP, it is not known rigorously that f(ℓ) is eventually monotonically increasing. Here, for the Poisson-based Euclidean FPP of Howard and Newman (Prob. Theory Relat. Fields 108 (1997), 153–170), we prove an explicit formula for f′(ℓ). In all dimensions, for certain values of the model's only parameter we have f′(ℓ) ≥ C > 0 for large ℓ.