On the time constant of high dimensional first passage percolation

We study the time constant $μ(e_{1})$ in first passage percolation on $\mathbb Z^{d}$ as a function of the dimension. We prove that if the passage times have finite mean, $\lim_{d \to \infty} \frac{μ(e_{1}) d}{\log d} = \frac{1}{2a},$ 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.

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