Singularity points for first passage percolation

Abstract

Let $0<a<b<\infty$ be fixed scalars. Assign independently to each edge in the lattice ${\mathbb{Z}}^2$ the value $a$ with probability $p$ or the value $b$ with probability $1-p$. For all $u,v\in {\mathbb{Z}}^2$, let $T(u,v)$ denote the first passage time between $u$ and $v$. We show that there are points $x\in {\mathbb{R}}^2$ such that the ``time constant'' in the direction of $x$, namely, ${\lim }_{n\to \infty }{n}^{-1}{\mathbf{E}}_p[T(0,nx)],$ is not a three times differentiable function of $p$.

Published at http://dx.doi.org/10.1214/009117905000000819 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)