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

Dominic Bair, Sagnik Jana, Yulan Qing

TL;DR

This work analyzes first-passage percolation (FPP) on infinite graphs with i.i.d. edge weights to understand how random metric perturbations affect global geometry. The authors prove that non-positive curvature properties—specifically Gromov hyperbolicity and coarse CAT(0) geometry—are almost surely not preserved under FPP, and that Morse geodesics lose their Morse character after perturbation. They also show FPP acts radially, providing precise large-scale Lipschitz and lower-bound bounds along geodesic rays from a basepoint. Collectively, the results clarify how random edge-length perturbations disrupt negative-curvature phenomena while still inducing a structured radial behavior, with implications for boundaries and Morse structures in random geometric settings.

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, $ν$, 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 $G$.

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

TL;DR

This work analyzes first-passage percolation (FPP) on infinite graphs with i.i.d. edge weights to understand how random metric perturbations affect global geometry. The authors prove that non-positive curvature properties—specifically Gromov hyperbolicity and coarse CAT(0) geometry—are almost surely not preserved under FPP, and that Morse geodesics lose their Morse character after perturbation. They also show FPP acts radially, providing precise large-scale Lipschitz and lower-bound bounds along geodesic rays from a basepoint. Collectively, the results clarify how random edge-length perturbations disrupt negative-curvature phenomena while still inducing a structured radial behavior, with implications for boundaries and Morse structures in random geometric settings.

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, a process called first-passage percolation. Assume that the graph is infinite and of bounded degree. Assume the edge length distribution, , has a finite expectation and is supported on . 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 .

Paper Structure

This paper contains 12 sections, 23 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. preliminaries
  3. First-passage percolation
  4. Geometric implication of first-passage percolation
  5. Basic probabilistic tools
  6. Background in coarse geometry
  7. Cayley graphs and group actions
  8. First-passage percolation for hyperbolic graphs
  9. Proof of Theorem \ref{['introthm1']}
  10. Coarse $\operatorname{CAT}(0)$ spaces and first passage percolations
  11. First-passage percolation and Morse sets
  12. First passage percolation is a radial map

Key Result

Theorem A

Let $X$ be a Gromov hyperbolic space, and an infinite, connected graph with bounded degree. Suppose for any bounded set $B$, at least one component of $X {\smallsetminus} B$ is not a tree. Assume $\mathbb{E} \omega_e < \infty$, $\text{Supp}(\nu) = (0,\infty)$, and $\nu(\{0\}) = 0$. Then, there exist

Figures (3)

  • Figure 1: BH1 Non-fat triangles: The distance between the points x and y is no greater than the distance between their comparison points $\overline x$ and $\overline y$.
  • Figure 2: $\text{Cay}(\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/3\mathbb{Z}, \langle a,b ~|~ a^2,b^3\rangle)$.
  • Figure 3: Diagram showing that $T_n$ having side $\omega$-lengths $3\delta$ is not $\delta$-hyperbolic, but is an $\omega$--triangle.

Theorems & Definitions (53)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D: Corollary \ref{['cor:radial']}
  • Definition 2.1
  • Definition 2.2: HW65, 2.2
  • Theorem 2.3: HW65, Theorem 2.2.1
  • Example 2.4
  • Proposition 2.5: BT17, Lemma 2.3
  • Proposition 2.6: Lemma 2.4 BT17
  • ...and 43 more