Weakest edge-noise upper-tail condition

Determine the weakest upper-tail conditions on the edge-noise distribution that preserve the passage-time bounds obtained for the inhomogeneous long-range first-passage percolation model.

Background

The proofs assume only that the edge noise has a finite positive moment, and this moment condition is used to control minima of several independent edge noises and endpoint attachments. The authors note that the exponent appearing in the resulting operational bound is a sufficient condition arising from their construction rather than a fundamental restriction.

The unresolved issue is to identify the weakest integrability or upper-tail hypothesis under which the stated passage-time bounds continue to hold.

References

Therefore, the exponent \zeta(1-\beta) in Proposition~\ref{prop:lin-ub-input} is a sufficient operational bound derived from our construction, rather than a fundamental restriction. Determining the weakest upper-tail conditions that preserve these passage-time bounds remains an open question.

— Inhomogeneous Long-Range First-Passage Percolation in a Random Vertex Environment  (2609.20422 - Chatterjee et al., 17 Sep 2026) in Remark on the moment assumption, Section 1.4 (Discussion on the assumptions)