Distributional limit in Regime II

Determine whether the passage times \(T_n=T(0,\lceil n x\rceil)\) converge in distribution in Regime II, and identify the resulting limit law and the law of the almost-surely finite explosion time \(\tau_\infty\).

Background

In Regime II, the paper proves that $T_n$ is tight and that the metric has a finite explosion time almost surely. The hub-chain construction suggests that the limiting object may be related to the passage time along a bi-infinite chain of exceptionally heavy vertices.

A complete limit description would need to account simultaneously for the point process of large vertex weights, the edge noises, and the environments at the two endpoints. The authors explicitly leave both convergence in distribution and identification of the explosion-time law unresolved.