Lower bounds for fluctuations in first-passage percolation for general distributions

TL;DR

This paper establishes new lower bounds on the fluctuations of passage times in first-passage percolation for general distributions, showing that these fluctuations are significantly larger than previously proven, even in higher dimensions.

Contribution

It introduces methods to prove stronger fluctuation bounds for FPP with general edge-weight distributions, extending results beyond special cases.

Findings

Fluctuations of T(0,x) are with high probability not contained in small intervals.

Established lower bounds on fluctuations in thin cylinders.

Results apply to general edge-weight distributions, not just special cases.

Abstract

In first-passage percolation (FPP), one assigns i.i.d.~weights to the edges of the cubic lattice $\mathbb{Z}^d$ and analyzes the induced weighted graph metric. If $T(x,y)$ is the distance between vertices $x$ and $y$, then a primary question in the model is: what is the order of the fluctuations of $T(0,x)$? It is expected that the variance of $T(0,x)$ grows like the norm of $x$ to a power strictly less than 1, but the best lower bounds available are (only in two dimensions) of order $\log \|x\|$. This result was found in the '90s and there has not been any improvement since. In this paper, we address the problem of getting stronger fluctuation bounds: to show that $T(0,x)$ is with high probability not contained in an interval of size $o(\log \|x\|)^{1/2}$, and similar statements for FPP in thin cylinders. Such statements have been proved for special edge-weight distributions, and here we obtain such bounds for general edge-weight distributions. The methods involve inducing a fluctuation in the number of edges in a box whose weights are of "hi-mode" (large).