Sharp fluctuations and oscillations in Regime III

Determine whether \((\log n)^{-\Delta_{\mathrm{III}}}T_n\) is tight in Regime III and whether it exhibits asymptotic oscillations, potentially including \(\log\log n\)-periodic oscillations.

Background

For Regime III, the paper proves a polylogarithmic upper bound of order (log⁡n)ΔIII+(\log n)^{\Delta_{\mathrm{III}}+}, but does not establish matching sharp asymptotics. The binary multiscale construction has an integer-valued depth, which may create oscillatory behavior analogous to arithmetic oscillations in long-range percolation.

The authors specifically identify tightness of the normalized passage time and the existence of oscillations as unresolved, noting that proving periodic oscillations may require stronger tail assumptions than the logarithmic tail hypotheses used in the paper.

References

In Regime III, is (\log n){-\Delta_{\rm III}}T_n tight, and does it oscillate? In long-range percolation, arithmetic oscillations arise because the depth k of the relevant multiscale structure is an integer. The binary tree construction suggests a similar mechanism, but establishing \log\log n-periodic oscillations for the passage metric likely requires finer tail assumptions than A3--A4.

— Inhomogeneous Long-Range First-Passage Percolation in a Random Vertex Environment  (2609.20422 - Chatterjee et al., 17 Sep 2026) in Item 2, Subsection 1.5.1 (Sharp asymptotics)