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

TL;DR

The paper resolves a gap in high-dimensional first-passage percolation by establishing the precise asymptotics of the time constant $\mu_d(e_1)$ as $d \to \infty$, under near-zero behavior of the edge-weight distribution $F$. It combines Dhar's Eden-model exploration with a coupling to general i.i.d. weights and a second-moment analysis to obtain convergence in probability and in $L^1$, thereby proving $\lim_{d\to\infty} \mu_d(e_1) d/\log d = 1/(2a)$ and the corresponding upper bound for general $F$ via uniform integrability. A key auxiliary result shows the non-backtracking hyperplane passage time in the exponential case has variance $o((\log d/d)^2)$, and the methods extend the upper bound from exponential to broader weight laws. These findings clarify the time-constant behavior in high dimensions and solidify the connection between Eden-type exploration and general first-passage percolation.

Abstract

In [2], it was claimed that the time constant $μ_{d}(e_{1})$ for the first-passage percolation model on $\mathbb Z^{d}$ is $μ_{d}(e_{1}) \sim \log d/(2ad)$ as $d\to \infty$, if the passage times $(τ_{e})_{e\in \mathbb E^{d}}$ 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 $o\big((\log d/d)^{2}\big)$ as $d\to \infty$ in the case of the when the edge weights are exponentially distributed.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Proposition 1: AT16highdimenFPP
• Remark 1
• proof : Proof of Theorem \ref{['thm1']}
• Theorem 3
• Proposition 2
• proof : Proof of Proposition \ref{['conv:probaforS']}
• Proposition 3
• proof : Proof of Proposition \ref{['prop:lastone']}