Nondifferentiability of Time Constants for First-Passage Percolation

We study the paths of minimal cost for first-passage percolation in two dimensions and obtain an exponential bound on the tail probability of the ratio of the lengths of the shortest and longest of these. This inequality permits us to answer a long-standing question of Hammersley and Welsh (1965) on the shift differentiability of the time constant. Specifically, we show that for subcritical Bernoulli percolation the time constant is not shift differentiable when p is close to one-half.