Definition: External regularity assertions

Let ${\mu }_F$ be the time-constant norm for undirected nearest-neighbour first-passage percolation on ${\mathbb{Z}}^2$ with i.i.d.\ edge law $F$.

1. For $F=\mathrm{Exp}(\lambda )$, $\lambda >0$, the unit ball of ${\mu }_F$ is strictly convex, and ${\mu }_F$ is Fréchet differentiable on ${\mathbb{R}}^2\setminus \{0\}$.

2. For $F=\mathrm{Gamma}(\kappa ,\lambda )$, $\kappa ,\lambda >0$, ${\mu }_F$ is Fréchet differentiable on ${\mathbb{R}}^2\setminus \{0\}$.

These are conditional interfaces to the claims of 212-02, not theorems proved or adopted unconditionally in this paper.