Disjoint finite geodesics in first-passage percolation

TL;DR

This paper studies the existence and probability of disjoint geodesics in first-passage percolation on high-dimensional lattices, providing bounds under various distributional assumptions.

Contribution

It establishes new probabilistic bounds for disjoint geodesics in first-passage percolation, extending understanding of their occurrence in lattice models.

Findings

Lower bounds for probability of disjoint geodesics between vertices.

Results applicable to various distributions of passage times.

Insights into the structure of geodesics in high-dimensional lattices.

Abstract

We investigate first-passage percolation on the lattice $\Z^d$ for dimensions $d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $\tau$. We only assume $\P[\tau=0]0$ is explicit) for the probability of having two disjoint geodesics between two pairs of neighbouring vertices at distance $n$. Additionally, under more specific assumptions on the distribution of $\tau$, we obtain similar lower bounds for the probability of having two disjoint geodesics (except for their starting and ending points) between the same two vertices.