Singularity points for first passage percolation

الأصل · EN

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,v², let T(u,v) denote the first passage time between u and v. We show that there are points x² such that the ``time constant'' in the direction of x, namely, ₙ→∞n⁻¹Eₚ[T(0,nx)], is not a three times differentiable function of p.