临界Erdős-Rényi分量过程的一致高阶阶乘矩界
Uniform High Order Factorial Moment Bounds for the Critical Erdős-Rényi Component Process
AI总结：
该研究对Erdős-Rényi随机图的Aldous临界窗口极限的局部盈余标记点过程形式，给出有限n计数推导，得到一致高阶阶乘矩界，还推导了有序ℓ²分量大小收敛，为相关研究提供了关键矩估计。
AI中文摘要：
我们对Erdős-Rényi随机图的Aldous临界窗口极限的局部盈余标记点过程形式，给出了有限n计数推导。对于pₙ = n⁻¹ + λn⁻⁴/³，令Ξₙ在每个分量的重标度大小和盈余处放置一个原子。精确的分量计数给出了极限阶乘相关密度和一致界：E[(Ξₙ(K))ₚ] ≤ C_Kᵠ e⁻ᶜᴷᵠ³，q ≥ 1，对每个紧标记窗口K，在容许的n中一致成立。三次阶对极限阶乘测度是最优的，该估计在紧大小区间上对所有盈余求和后仍成立。它给出了过密和局部指数矩界、有限n拉普拉斯泛函展开的定量截断，以及局部点过程收敛。利用Janson-Spencer Palm描述和经典的全盈余估计，我们还得到了有序ℓ²分量大小收敛。
英文摘要：
We give a finite $n$ enumerative derivation of the local surplus marked point process form of Aldous's critical window limit for the Erdős-Rényi random graph. For $p_n=n^{-1}+λn^{-4/3}$, let $Ξ_n$ place an atom at the rescaled size and surplus of each component. Exact component enumeration yields the limiting factorial correlation densities and the uniform bound \[ \mathbb E[(Ξ_n(K))_q]\le C_K^q e^{-c_Kq^3}, \qquad q\ge1, \] for every compact marked window $K$, uniformly in admissible $n$. The cubic order is optimal for the limiting factorial measures, and the estimate persists after summing over all surpluses on compact size intervals. It yields overcrowding and local exponential moment bounds, quantitative truncation of the finite $n$ Laplace functional expansion, and local point process convergence. Using the Janson-Spencer Palm description and a classical all excess estimate, we also recover ordered $\ell^2$ component size convergence.
