Properties of first passage percolation above the (hypothetical) critical dimension

TL;DR

This paper investigates the hypothetical properties of first passage percolation in dimensions above the critical dimension, focusing on how certain fundamental properties would differ if the fluctuation exponent were zero.

Contribution

It analyzes the implications of a zero fluctuation exponent in high dimensions and suggests that certain known properties of FPP would fail, indicating a local nature of passage times.

Findings

At least one fundamental property of FPP must be false if the fluctuation exponent is zero.

Passage times become predominantly determined by local configurations in the zero fluctuation regime.

Disc-to-disc passage times are significantly faster than mean point-to-point passage times in this regime.

Abstract

It is not known (and even physicists disagree) whether first passage percolation (FPP) on $\mathbb{Z}^d$ has an upper critical dimension $d_c$, such that the fluctuation exponent $\chi=0$ in dimensions $d>d_c$. In part to facilitate study of this question, we may nonetheless try to understand properties of FPP in such dimensions should they exist, in particular how they should differ from $d<d_c$. We show that at least one of three fundamental properties of FPP known or believed to hold when $\chi>0$ must be false if $\chi=0$. A particular one of the three is most plausible to fail, and we explore the consequences if it is indeed false. These consequences support the idea that when $\chi=0$, passage times are ``local'' in the sense that the passage time from $x$ to $y$ is primarily determined by the configuration near $x$ and $y$. Such locality is manifested by certain ``disc--to--disc'' passage times, between discs in parallel hyperplanes, being typically much faster than the fastest mean passage time between points in the two discs.