Lower bounds on non-random fluctuations in planar first passage percolation

TL;DR

This paper establishes a lower bound on the non-random fluctuations in planar first passage percolation, showing they grow at least as fast as a power of log(n), which advances understanding of fluctuation divergence.

Contribution

It improves previous bounds on fluctuations in planar FPP and demonstrates divergence for arbitrary directions, using recent theoretical developments.

Findings

Non-random fluctuations grow at least as log(n)^α for α<1/2.

First divergence result for arbitrary directions in planar FPP.

Improves previous log(log(n)) bound by Nakajima.

Abstract

The fluctuations of the passage time in first passage percolation are of great interest. We show that the non-random fluctuations in planar FPP are at least of order $\log(n)^\alpha$ for any $\alpha<1/2$ under some conditions that are known to be met for a large class of absolutely continuous edge weight distributions. This improves the ${\log(\log(n))}$ bound proven by Nakajima and is the first result showing divergence of the fluctuations for arbitrary directions. Our proof is an application of recent work by Dembin, Elboim and Peled on the BKS midpoint problem and the development of Mermin-Wagner type estimates.