On the time constant of high dimensional first passage percolation

TL;DR

This paper investigates how the time constant in high-dimensional first passage percolation behaves as the dimension grows, revealing a precise asymptotic relation and the shape of the limit set.

Contribution

It establishes the asymptotic behavior of the time constant in high dimensions and characterizes the limit shape for a broad class of passage time distributions.

Findings

The time constant scaled by dimension and log d converges to 1/(2a).

The limit shape is not Euclidean ball, cube, or diamond in high dimensions.

The asymptotic behavior depends only on the distribution's behavior at zero.

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, $$\lim_{d \to \infty} \frac{\mu(e_{1}) d}{\log d} = \frac{1}{2a},$$ 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.