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, $$\underset{d\to \infty }{\lim }\frac{\mu (e_1)d}{\log d}=\frac{1}{2a},$$ 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.

25 pages, 5 figures