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
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.
@article{1048516544,
author = {Steele, J. Michael and Zhang, Yu},
title = {Nondifferentiability of the time constants of first-passage percolation},
journal = {Ann. Probab.},
volume = {31},
number = {1},
year = {2003},
pages = { 1028-1051},
language = {en},
url = {http://dml.mathdoc.fr/item/1048516544}
}
Steele, J. Michael; Zhang, Yu. Nondifferentiability of the time constants of first-passage percolation. Ann. Probab., Tome 31 (2003) no. 1, pp. 1028-1051. http://gdmltest.u-ga.fr/item/1048516544/