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

TL;DR

This paper revisits the asymptotic behavior of the time constant in high-dimensional first passage percolation, correcting a previous proof and establishing new bounds, with implications for variance estimates in exponential cases.

Contribution

It provides a new approach to establish the upper bound of the time constant's asymptotics and analyzes variance in the exponential case, correcting prior inaccuracies.

Findings

Established the upper bound of the time constant in high dimensions.

Proved the variance of the passage time is of smaller order in exponential cases.

Corrected the proof of a key asymptotic inequality in first passage percolation.

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{\mu_{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 $o\big((\log d/d)^{2}\big)$ as $d\to \infty$ in the case of the when the edge weights are exponentially distributed.