Divergence of non-random fluctuation in First Passage Percolation

TL;DR

This paper proves that non-random fluctuations in first passage percolation on integer lattices diverge in all dimensions, confirming a longstanding conjecture about shape fluctuation divergence.

Contribution

It establishes the divergence of non-random fluctuations and shape fluctuations in first passage percolation, extending previous conjectures to all dimensions.

Findings

Non-random fluctuations diverge in all dimensions

Shape fluctuation divergence is confirmed

Supports conjecture by Yu Zhang (2006)

Abstract

We study non-random fluctuation in the first passage percolation on $\mathbb{Z}^d$ and show that it diverges for any dimension. We also prove the divergence of the non-random shape fluctuation, which was conjectured in [Yu Zhang. The divergence of fluctuations for shape in first passage percolation. {\em Probab. Theory. Related. Fields.} 136(2) 298--320, 2006].