AI 中文总结
本文针对单模随机树上阈值级联，提出多尺度前端解耦准则，通过重整化群分析证明在谱半径条件下解耦成立，并扩展了有效耗散条件，揭示了共振区的相边界。
AI 中文摘要
这是关于耦合Ornstein-Uhlenbeck扩散在图上阈值级联的一系列论文中的第三篇，该图在Benjamini-Schramm意义下收敛到单模Galton-Watson树。在arXiv:2608.15987中，级联到有限型Galton-Watson过程的约简（前端解耦）在耗散区域κ<1下被无条件证明，其中κ=γLσΔmax/αmin是单步耗散比。当κ≥1时，单步回波界共振，该约简仍为猜想。我们将其替换为多尺度（重整化群）分析。将空间-时间块粗粒化为深度L和持续时间T的块，我们记录每个块从边界到边界的增益，形成一个2×2矩阵，包含轨迹通道和存储质量通道，扇出被吸收。由此得到三个定理。增益矩阵在深度和时间拼接下可复合，因此存在重整化的每层耗散比κ_eff，且等于其下确界（Fekete）。若增益矩阵的谱半径在单一有限尺度下小于1（原则上可通过数值验证的有限时域、有限体积条件），则前端解耦在所有尺度上成立，并具有定量的全变差界；回波可局部共振，但宏观上恢复独立性。最后，我们将影响步骤分为前端、静止和饱和阶段，根据其源的适应频带占用而非全局最坏情况为每一步定价，并推导出κ_eff≤ρ(T)，其中T为显式信道矩阵。区域ρ(T)<1严格包含κ<1：接近临界时，有效条件从γLσΔmax/α<1退化为γLσ/α<1，且度消失。我们未证明在κ≥1全域内κ_eff<1，并给出共振区应为真正相边界的启发式论证。
英文摘要
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.
