单模随机树上交互扩散的阈值级联的两个问题：带Bramson修正的前沿传播与连续型极限理论
Two problems for threshold cascades of interacting diffusions on unimodular random trees: front propagation with a Bramson correction, and the continuous-type limit theory
浏览论文内容
中文总结 AI 辅助
本文针对单模随机树上交互扩散阈值级联的两个未解决问题，提供分析框架并证明部分定理，涉及前沿传播与连续型极限理论，给出相关猜想与证明策略，未完全解决两个问题。
中文摘要 AI 辅助
本文是arXiv:该链接的配套论文，该论文将图上耦合Ornstein-Uhlenbeck扩散的阈值级联在Benjamini-Schramm收敛到单模Galton-Watson树的耗散区域内无条件地约化为有限型Galton-Watson过程。该配套论文中留有两个未解决的问题，本文对二者进行精确表述，提供分析框架并证明部分定理。第一个问题是前沿传播：本文证明按代索引的前沿存在真实的分支随机游走比较，证明前沿深度以弹道式增长，其显式速度c*由倾斜均值矩阵的Perron根给出（在耗散区域内无条件成立），并将猜想的Bramson延迟c*t - (3/(2c*))log t约化为导数鞅的一致可积性，本文在线性化层面构造了该导数鞅；前沿中心极限定理和Bramson修正作为猜想给出，并附有证明策略。第二个问题是敏感区域，其中类型为连续失效强度且后代分布为Cox混合：本文证明强度变量的L²上的均值后代算子K是拟紧的且具有谱间隙，其机制是控制强度遗传的Ornstein-Uhlenbeck核的Hilbert-Schmidt平滑；这得到了一般状态空间的Kesten-Stigum定理，将n^{-3/2}的总后代律扩展到带有显式常数的连续型，并给出了强度分辨的中心极限定理；连续型CRT标度极限被约化为Polish型空间上的多型不变性原理，该猜想的所有假设除一个紧性估计外均已验证。两个问题均未完全解决，每个问题都提供了框架、一阶定理和明确界定的剩余步骤。
英文摘要
Companion to arXiv:2608.XXXXX, which reduces threshold cascades of coupled Ornstein-Uhlenbeck diffusions on graphs converging Benjamini-Schramm to a unimodular Galton-Watson tree, unconditionally in a dissipative regime, to a finite-type Galton-Watson process. Two problems are left open there; we formulate both precisely, supply the analytic framework, and prove partial theorems. First, front propagation. We show the generation-indexed front admits a genuine branching random walk comparison, prove the front depth grows ballistically with an explicit speed $c_*$ given by the Perron root of a tilted mean matrix (unconditional in the dissipative regime), and reduce the conjectured Bramson delay $c_* t - \frac{3}{2c_*}\log t$ to the uniform integrability of a derivative martingale, which we construct at the linearised level. The front central limit theorem and the Bramson correction are stated as conjectures with a proof strategy. Second, the sensitive regime, where the type is a continuous failure strength and the offspring law is a Cox mixture. We prove the mean offspring operator $K$ on $L^2$ of the strength variable is quasi-compact with a spectral gap, the mechanism being Hilbert-Schmidt smoothing of the Ornstein-Uhlenbeck kernel governing strength inheritance. This yields a general-state-space Kesten-Stigum theorem, extends the $n^{-3/2}$ total-progeny law to continuous types with an explicit constant, and gives a strength-resolved central limit theorem. The continuous-type CRT scaling limit is reduced to a multitype invariance principle on a Polish type space, conjectured with all hypotheses verified modulo one tightness estimate. Neither problem is fully closed; each is given a framework, first-order theorems, and a precisely delimited remaining step.
