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

Abstract

In first-passage percolation (FPP) models, the passage timeTℓfrom the origin to the pointℓeℓsatisfiesf(ℓ) := ETℓ= μℓ+o(ℓ½+ε), where μ ∊ (0,∞) denotes the time constant. Yet, for lattice FPP, it is not known rigorously thatf(ℓ) is eventually monotonically increasing. Here, for the Poisson-based Euclidean FPP of Howard and Newman (Prob. Theory Relat. Fields108(1997), 153–170), we prove an explicit formula forf′(ℓ). In all dimensions, for certain values of the model's only parameter we havef′(ℓ) ≥C> 0 for largeℓ.