If an infinite bounded-degree graph has a sublinearly Morse bi-infinite quasi-geodesic, then first passage percolation almost surely has a bi-infinite geodesic.
A Characterization of Higher Rank Symmetric Spaces Via Bounded Cohomology , Geometric and Functional Analysis volume 19, pages 11--40 (2009) comment
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Sublinear Morse Geodesics and First Passage Percolation
If an infinite bounded-degree graph has a sublinearly Morse bi-infinite quasi-geodesic, then first passage percolation almost surely has a bi-infinite geodesic.