Sublinear Morse Geodesics and First Passage Percolation

TL;DR

This paper proves that in infinite graphs with randomly perturbed edge lengths, the existence of a sublinearly Morse bi-infinite geodesic implies the almost sure existence of a bi-infinite geodesic line, generalizing previous results.

Contribution

It extends the understanding of Morse geodesics in randomly perturbed graphs by establishing the almost sure existence of bi-infinite geodesic lines under broader conditions.

Findings

Almost surely existence of bi-infinite geodesic lines in perturbed graphs.

Generalization of previous Morse geodesic results.

Applicability to graphs with bounded degree and random edge lengths.

Abstract

Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges of the graph. Assume that the graph is infinite and of bounded degree. Assume also strict positivity and finite expectation of the edge length distribution and existence of a sublinearly Morse bi-infinite geodesic line, we prove that almost surely there exists a bi-infinite geodesic line. This generalizes a previous result of \cite{BT17} regarding Morse geodesics.