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

Abstract

In first-passage percolation (FPP) models, the passage time T_l from the origin to the point le₁ 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.