A multiscale front-decoupling criterion for threshold cascades on unimodular random trees

Abstract

This is the third paper in a series on threshold cascades of coupled Ornstein-Uhlenbeck diffusions on graphs converging Benjamini-Schramm to a unimodular Galton-Watson tree. In arXiv:2608.15987 the reduction of the cascade to a finite-type Galton-Watson process (front decoupling) was proved unconditionally in the dissipative regime kappa < 1, where kappa = gamma L_sigma Delta_max / alpha_min is the single-step dissipation ratio. At kappa >= 1 the single-step echo bound resonates, and the reduction was left conjectural. We replace it by a multiscale (renormalisation-group) analysis. Coarse-graining into space-time blocks of depth L and duration T, we record each block's boundary-to-boundary gains in a 2x2 matrix with a trajectory channel and a stored-mass channel, fan-out absorbed. Three theorems result. The gain matrices compose under depth and time concatenation, so a renormalised per-level dissipation ratio kappa_eff exists and equals its infimum (Fekete). If the spectral radius of the gain matrix is below 1 at a single finite scale, a finite-horizon, finite-volume condition in principle certifiable numerically, then front decoupling holds at all scales with a quantitative total-variation bound; echoes may resonate locally, yet independence is recovered macroscopically. Finally we split influence steps into front, quiescent and saturated phases, price each step by the adapted band occupation of its source rather than by a global worst case, and deduce kappa_eff <= rho(T) for an explicit channel matrix T. The region rho(T) < 1 strictly contains kappa < 1: near criticality the effective condition degrades from gamma L_sigma Delta_max / alpha < 1 to gamma L_sigma / alpha < 1, and the degree disappears. We do not prove kappa_eff < 1 throughout kappa >= 1, and give a heuristic for why the resonance regime should be a genuine phase boundary.