2025/12/05 by Dominic Bair, Bair, Dominic, S. Jana +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2512.05720
openalex publication_date 2025/12/05 · openalex created_date 2025/12/09 · openalex updated_date 2026/07/28
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, ν, has a finite expectation and is supported on [0, ∞). 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.