First passage percolation preserves sublinearly Morse boundaries

TL;DR

This paper demonstrates that sublinearly Morse boundaries in proper geodesic spaces remain invariant under first passage percolation, providing an alternative characterization and studying their stability in random graph models.

Contribution

It introduces an alternative characterization of sublinearly Morse geodesics via middle recurrence and proves their invariance under FPP on proper geodesic graphs.

Findings

Sublinearly Morse boundaries are invariant under FPP.

Provides an alternative characterization of sublinearly Morse geodesics.

Establishes stability of these boundaries in random graph models.

Abstract

Sublinearly Morse directions in proper geodesic spaces are defined by sublinearly Morse stability. In this paper we offer an alternative characterization for sublinearly Morse geodesic lines via middle recurrence. We then study first passage percolation (FPP) on proper geodesic graphs of bounded degree. We associate an i.i.d. collection of random passage times to each edge. Under suitable conditions on the passage time distribution, we prove that sublinearly Morse boundaries are invariant under first passage percolation.