On the time constant of high dimensional first passage percolation, revisited

Abstract

In [2], it was claimed that the time constant ${\mu }_d(e_1)$ for the first-passage percolation model on ${\mathbb{Z}}^d$ is ${\mu }_d(e_1)\sim \log d/(2ad)$ as $d\to \infty$, if the passage times $({\tau }_e{)}_{e\in {\mathbb{E}}^d}$ are i.i.d., with a common c.d.f. $F$ satisfying $\left|\frac{F(x)}{x}-a\right|\le \frac{C}{|\log x|}$ for some constants $a,C$ and sufficiently small $x$. However, the proof of the upper bound, namely, Equation (2.1) in [2] \begin{align} \limsup_{d\to\infty} \frac{μ_{d}(e_{1})ad}{\log d} \le \frac{1}{2} \end{align} is incorrect. In this article, we provide a different approach that establishes this inequality. As a side product of this new method, we also show that the variance of the non-backtracking passage time to the first hyperplane is of order $o((\log d/d{)}^2)$ as $d\to \infty$ in the case of the when the edge weights are exponentially distributed.

14 pages