On properties of optimal paths in first passage percolation

TL;DR

This paper investigates properties of optimal paths in first passage percolation on ^d, revealing exponential growth in the number of optimal paths with atoms in the distribution and linearity in intersection and union means.

Contribution

It introduces new insights into the structure of optimal paths, including exponential growth conditions and linearity of intersection and union means, using a configuration-flipping argument.

Findings

Number of optimal paths grows exponentially with an atom in the distribution.

Means of intersection and union of optimal paths are linear in distance.

Utilizes a configuration-flipping argument adapted from previous work.

Abstract

In this paper, we study some properties of optimal paths in the first passage percolation on $\Z^d$ and show the followings: (1) the number of optimal paths has an exponential growth if the distribution has an atom; (2) the means of intersection and union of optimal paths are linear in the distance. For the proofs, we use the configuration--flipping argument introduced in [J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 56-80, 1993] with suitable adaptions.