Singularity points for first passage percolation

Abstract

Let 0<a<b<∞ be fixed scalars. Assign independently to each edge in the lattice ℤ² the value a with probability p or the value b with probability 1−p. For all u,v∈ℤ², let T(u,v) denote the first passage time between u and v. We show that there are points x∈ℝ² such that the “time constant” in the direction of x, namely, lim _n→∞n⁻¹E_p[T(0,nx)], is not a three times differentiable function of p.