Front-Markov reduction in the full saturated regime

Establish the front-Markov reduction for the interacting diffusion cascade throughout the full saturated regime, including the strong-coupling regime with dissipation ratio \(\kappa\ge1\), uniformly through the critical window, and for unbounded-degree unimodular tree limits; equivalently, prove that the failed-cluster law is asymptotically that of the multitype Galton–Watson process with mean matrix \(M\), with conditional independence of successive generations given the current failed vertices and their types.

Background

The paper reduces the interacting diffusion cascade to a finite-type Galton–Watson genealogy under a front-decoupling hypothesis. This reduction is proved on the Bethe lattice and unconditionally in a dissipative region where $\kappa<1$ and echo decay dominates cluster growth. The unresolved case is the remainder of the saturated regime, particularly strong coupling, where the weighted-Grönwall contraction used to control back-reaction echoes is unavailable.

The authors also identify two related uniformity issues: obtaining estimates through the critical window without requiring the joint scaling condition $n\varepsilon(\delta_n)\to0$, and extending the front-decoupling argument beyond bounded degrees. Resolving this problem would make the branching-process description valid for the full saturated model rather than only conditionally or in restricted parameter regimes.