On the time constant of high dimensional first passage percolation

Abstract

We study the time constant $\mu (e_{1})$ in first passage percolation on $\mathbb Z^{d}$ as a function of the dimension. We prove that if the passage times have finite mean, $$\lim_{d\to\infty}\frac{\mu({e}_{1})d}{\log d} = \frac{1}{2a},$$ where $a \in [0,\infty ]$ is a constant that depends only on the behavior of the distribution of the passage times at $0$. For the same class of distributions, we also prove that the limit shape is not an Euclidean ball, nor a $d$-dimensional cube or diamond, provided that $d$ is large enough.