Divergence of non-random fluctuation for Euclidean first-passage percolation

Abstract

The non-random fluctuation is one of the central objects in first passage percolation. It was proved in [11] that for a particular asymptotic direction, it diverges in a lattice first passage percolation with an explicit lower bound. In this paper, we discuss the non-random fluctuation in Euclidean first passage percolations and show that it diverges in dimension $d \geq 2$ in this model also. Compared with the result in [11], the present result is proved for any direction and improves the lower bound.