Finite-torus passage-time and flooding asymptotics

Determine the normalizations and limiting laws, as the torus size \(n\to\infty\), for the passage time between two uniformly sampled vertices, the flooding time, and the diameter of finite-torus inhomogeneous long-range first-passage percolation.

Background

The paper defines the finite-torus version of the model on Tnd\mathbb{T}_n^d, including power-law and finite-range cost functions, but does not analyze its global metric observables.

The unresolved finite-volume problems concern typical passage times between uniformly sampled vertices as well as the more extreme flooding-time and diameter observables, for which different tail assumptions may be necessary.

References

Determine the normalizations and limiting laws of the passage time between two uniformly sampled vertices, the flooding time, and the diameter as n\to\infty.

— Inhomogeneous Long-Range First-Passage Percolation in a Random Vertex Environment  (2609.20422 - Chatterjee et al., 17 Sep 2026) in Subsection 1.5.4 (Finite \(d\)-dimensional tori and other cost functions)

Which additional tail assumptions are needed for the flooding time and diameter, as opposed to a typical passage time?

— Inhomogeneous Long-Range First-Passage Percolation in a Random Vertex Environment  (2609.20422 - Chatterjee et al., 17 Sep 2026) in Subsection 1.5.4 (Finite \(d\)-dimensional tori and other cost functions)