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

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 $\sqrt{\log n}$ 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.