First-passage percolation, non-positive curvature, and radial maps

TL;DR

This paper investigates how random perturbations of graph metrics via first-passage percolation affect geometric properties like non-positive curvature, hyperbolicity, and Morse geodesics, showing these are generally not preserved.

Contribution

It proves that non-positive curvature, Gromov hyperbolicity, and coarse CAT(0) properties are almost surely not preserved under first-passage percolation, and that the perturbed map is radial.

Findings

Non-positive curvature is almost surely not preserved.

Gromov hyperbolicity is almost surely not preserved.

First-passage percolation is almost surely a radial map.

Abstract

Given an infinite connected graph $G$, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges of the graph, a process called first-passage percolation. Assume that the graph is infinite and of bounded degree. Assume the edge length distribution, $\nu$, has a finite expectation and is supported on $[0, \infty)$. We prove in this paper that non-positive curvature almost surely is not preserved by the associated percolation. In particular, Gromov hyperbolicity and coarse CAT(0) property of graphs are almost surely not preserved. We also show that if a graph contains a Morse geodesic ray, then the resulting image of the ray under first-passage percolation is no longer Morse. Lastly, we show that first-passage percolation almost surely is a radial map on $G$.