单模随机树上相互作用扩散的阈值级联：条件分支过程普适性定理、退火压力与失效边界的维数
Threshold cascades of interacting diffusions on unimodular random trees: a conditional branching-process universality theorem, the annealed pressure, and the dimension of the failed boundary
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中文总结 AI 辅助
该研究在单模随机树上的耦合扩散系统中，证明临界性条件、失效簇边界维数等结果，否定部分自然论断，给出耗散 regime 下的简化证明，强耦合 regime 留作猜想。
中文摘要 AI 辅助
我们研究在有限图上的耦合Ornstein-Uhlenbeck扩散系统，该系统带有吸收失效态，且有限图在局部弱收敛到单模Galton-Watson树。单个失效可触发阈值穿越的级联，核心问题是该级联何时为亚临界或超临界，以及临界态附近的簇具有何种形态。在明确表述的前层面解耦假设（饱和耦合 regime）下，该假设在Bethe格上得到严格验证，根节点的失效簇依律收敛到有限型Galton-Watson过程的谱系，其均值矩阵M由单个矩阵值多对一公式计算得到。基于该简化，我们证明临界性发生在ρ(M)=1处，退火和淬火增长率为单个凸实解析压力φ(β)=logρ(M,β)在0处的切线和在1处的值，使得淬火/退火间隙满足弦-切线不等式；在存活条件下，失效簇的端边界具有Hausdorff维数logρ(M)；在临界态下，P(|S|=n)~C n^{-3/2}，且在矩假设下，由n^{-1/2}重标后的条件簇收敛到Aldous的连续随机树；亚临界域是开且可缩的，但非凸，谱隙1-ρ(M)给出定量稳定性。我们证明该设定中的若干自然论断不成立，包括Hilbert-Schmidt/Krein-Rutman算子方案、标量多对一恒等式以及谱半径/维数恒等式。该简化本身在耗散 regime κ<1时被无条件证明，其中路径式前向跟踪论证表明，反作用回波在图距离上呈几何衰减，且快于失效簇的体积增长；强耦合 regime 仍未解决，作为猜想提出。
英文摘要
We study systems of coupled Ornstein-Uhlenbeck diffusions with an absorbing failure state on finite graphs that converge locally weakly to a unimodular Galton-Watson tree. A single failure can trigger a cascade of threshold crossings, and the central question is when the cascade is subcritical or supercritical and what the cluster looks like near criticality. Under an explicitly stated front-decoupling hypothesis (the saturated-coupling regime), verified exactly on the Bethe lattice, the failed cluster of the root converges in law to the genealogy of a finite-type Galton-Watson process whose mean matrix M is computed from a single matrix-valued many-to-one formula. Conditional on this reduction we prove that criticality occurs at rho(M) = 1, that the quenched and annealed growth rates are the tangent at 0 and the value at 1 of a single convex real-analytic pressure phi(beta) = log rho(M(beta)), making the quenched/annealed gap a chord-tangent inequality, that on survival the end boundary of the failed cluster has Hausdorff dimension log rho(M), that at criticality P(|S| = n) ~ C n^{-3/2} and the conditioned cluster rescaled by n^{-1/2} converges to Aldous' Continuum Random Tree under a moment hypothesis, and that the subcritical domain is open and contractible but not convex, with the spectral gap 1 - rho(M) giving quantitative stability. Several natural claims in this setting are shown to be false, including the Hilbert-Schmidt/Krein-Rutman operator programme, the scalar many-to-one identity, and the spectral-radius/dimension identity. The reduction itself is proved unconditionally in the dissipative regime kappa < 1, where a pathwise front-tracking argument shows that back-reaction echoes decay geometrically in graph distance and beat the volume growth of the failed cluster; the strong-coupling regime remains open and is stated as a conjecture.
